jer2911mom,
This is a good question, and one that I haven't had time to thoroughly answer before now!

First off, it's important to note that in order to think on a higher level, kiddos first need some basic knowledge about the Bible, history, science, reading, math, the flow of time, etc. So, this is an important consideration in the younger years. It is very important to lay this foundation of basic knowledge prior to jumping into forcing kiddos to come up with their own conclusions and answers.
Within LHTH and LHFHG, we strive to lay a good foundation in all subject areas. This is the most important consideration in the younger years. We continue to build on this foundation throughout the coming guides.
As kiddos move into Beyond we begin broadening kiddos' horizons through reading and examining classic poetry, discovering the point of various Scriptures, composing oral narrations, and through deeper discussion of literature through the storytime part of the plans (and later more in-depth in Drawn into the Heart of Reading). Each of these areas begin moving a child's thinking beyond one right answer toward thinking more deeply about what was read. The Emerging Reader Set also has questions with the level of Bloom's Taxonomy noted behind each one, so you can easily see where the higher levels of thinking are coming in there! The Bible portion at times does lend itself toward one-right answers from Scripture, but that is to be expected as the Scripture does clearly set forth a right answer in God's eyes.
As kiddos move on to Bigger Hearts, we add more application and opportunities to develop higher-level thinking through DITHR and its open-ended questions asking the child to delve beneath the surface of what was read and make connections. The additonal literature study through the storytime box of plans is mainly higher level as well (i.e. as it applies discussion of Godly character traits, story elements, and genre study lessons to the readings). We also add oral narration practice to science as well as history allowing students to elaborate on what was read. Higher level questioning applies Scripture to history for students to use inference to draw conclusions as well as having student share connections to their own lives through the study of Godly character traits. Science Lab Forms expect students to make their own guesses, test them, and propose their own conclusions. We also ask kiddos to prove their understanding of vocabulary words by using them within their own sentences each week. A a deeper study of classic poetry and poetic devices has students explain the meaning of the poems and the way the poetic devices were used.
In Preparing Hearts, as kiddos gain a firmer founding in knowledge, they move toward more critical thinking. The Reading About History box labels questions according to Bloom's Taxonomy to make sure all levels of thinking are addressed. Written narrations lead students to compose their own thoughts in a written form and share their own connections, as they sift and sort through what to share from the readings. Storytime narrations have the children "imagine they are" the character and narrate in that style. They also discuss the character's faith (or lack of faith) and appraise the role it may have played in the story. Students are also asked to share something from their own life that they were reminded of during the readings, and initiate a discussion on their own topic that the reading brought to mind. The writing lessons based on the poetry of Robert Louis Stevenson go far beyond one right answer, as the students must compose their own poetry based on his style. The research component has students utilizing resources to focus on assigned topics and select what to share orally about what they've learned. Weekly science questions are included, some of which are always higher level. Both the Bible study and the Poetry study also include some higher level thought-provoking questions each week.
As students move into CTC, research of places and people continues. Students must elaborate on what was learned as they share their findings with a parent. Students move into giving detailed narrations one day and summary narrations the next, both of which require the student to process what was read, determine what to share, and support their connections. The Genesis study is definitely on a higher level thinking plane, as are the assignments for the geography. Students choose a memorable passage to include in their notebook for history each week, create their own timeline drawings each week, continue with deciding what to include within written narrations, choose connections in Proverbs to go with what they are reading in Storytime each week, and continue making their own guesses and drawing their own conclusions in their science lab sheets. Drawn into the Heart of Reading also raises the thinking to a higher level with the next level student book. Write with the Best pushes students beyond basic writing through asking them to model their writing after great passages of classic literature. Students must select their own topics while still including the needed guidelines.
When moving into RTR, research shifts to being about places read about in history. The research answers are integrated into a weekly postcard where students explain their responses to another person. Oral and written narration continue to play a big role in higher level thinking and connections. The Bible study of Philippians is an inductive study requiring students to interpret the meaning of the Biblical text and draw conclusions based on the surrounding text's meaning. The daily prayers require kiddos to meditate on the prayer topic and share from their own lives and hearts. The devotional Bible studies for boys and girls also have many great higher-level thinking moments within them as students plan to apply the Bible to their life. The science lab sheets and weekly written narration require higher thinking at these levels as well. The Shakespeare study is definitely one that challenges kiddos beyond their comfort zone. The poetry of Emily Dickinson also requires quite a bit of higher level thinking, simply to interpret what she meant! DITHR continues to raise the level further as kiddos are asked to dig more deeply into what was read.

Our math choice of Singapore Math at all levels is also heavily focused on higher level thinking!
Anyway, you get the idea that including higher-level thinking within our curriculum is something that we actively seek to do in each and every guide! These are certainly not all the instances of higher level thinking, but they are the ones that spring to mind.
One thing to be warned of is that while the box-like look of our daily plans or the scripted parts within our plans may lead you to think that higher level thinking is not contained in our guides, it is important to note that is just appearance. Higher-level questions often push toward higher level thinking and without the questions being included, we often wouldn't ask them on our own! After using our guides a year or more, you'll find they are meaty in many areas of higher-level thinking that often do not become apparent until you have used our guides as written for at least a year.
We realize that simply reading classic books does not lead to higher level thinking automatically, but rather it is what you do with what you are reading that makes the difference! After 11 years in as a classroom teacher, as well as having my master's in education, I can say that I was required to write lesson plans for every day of the school year that reflected higher-level thinking within the plans in some way. During my 10 years as a home educator since then, I strive daily for that same balance within HOD, knowing that some subjects are more knowledge type subjects and others are more inspirational type subjects, and we seek to balance both.

Higher level thinking can take many forms, and there is not one right way to include this in a curriculum, however it is important to look at this area in a way that glorifies God and not man. The greatest thinkers are still lost without a Savior.
Blessings,
Carrie