In most embedded questions, the "Wh" portion of the question stays grouped together with the rest of the converted question
Example:
"What time is it?" makes "Do you know what time it is?"
However, the rules seem different when "do you think" is used
Example:
"What time is it" makes "What time do you think it is?"
Is this splitting of the embedded question with "do you think" a unique exception?
There's something you missed. The two embedded questions are not parallel; one is a single clause, and the other (the one with think) is complex -- i.e, there are two clauses here instead of one.
Let's just look at the embedded question parts and see where they come from:
.. what time it is?
[indef] time -- by Wh-Question Formation (with what for [indef])[indef] -- by Extraposition (leaving a Dummy It subject)... what time you think it is?
[indef] time.[indef].Notice that in (2), the order is always it is, never is it. That's because the main clause that the wh-question is formed from is You think ..., rather than It is ... That's why the ordinary (not embedded) wh-question would need Do-Support, which of course is not the case in embedded questions.
So nothing has been "split". Syntax doesn't work that way. The rule has been applied to the main clause (with verb think) of a complex sentence, while the wh-word has been moved from the object complement clause (with verb is) of think.
Short answer. Tl;dr:
Yes/no questions just require an auxiliary verb to appear at the front of the sentence. This leaves the rest of the sentence in tact. This is what happened in the Original Poster's Do example.
Open questions require the interrogative word or phrase to be moved to the front of the sentence (when it is not the subject). This disrupts the rest of the sentence and can break up constituents. This is what has happened in the Original Poster's example with What time.
The full story:
Do you know what time it is?
What time do you think it is?
Example (1) is a yes/no question (also called a polar question or a closed question). To make a polar question we take the auxiliary verb and move it in front of the subject:
Sometimes a sentence doesn't have an auxiliary verb as standard in the declarative version. In these cases we use the dummy auxiliary, the verb DO:
This is what has happened in example (1):
Notice that although the speaker of such a sentence actually wants the listener to tell them what time it is, the technical answer to their question is either 'Yes' or 'No'. And if the listener wanted to be facetious they could easily give one of those answers as a very annoying joke.
Example (2) is rather different. It's an open question. The technical answer to (2) is not 'Yes' or 'No', but rather '2.15' or '3am' or whatever the time happens to be. It called an 'open' question because there is in principle no limit to the number of different answers one might give.
To make an open interrogative, we first need to replace the item we want to know the identity/value of with an interrogative word, like who or why and so forth:
Then if we want to make a conventional open question, we move the interrogative word or phrase to the front of the question:
But, as you can see from the examples above, that is not always sufficient. The second example, where the interrogative word is the subject is fine. However the first example—where we had to move the interrogative word from a different position to get it to the front of the sentence—is ungrammatical. In these cases, we also need to move the auxiliary verb to in front of the subject. And just like with yes/no questions, if we don't already have an auxiliary, we insert the dummy auxiliary, DO:
This is what has happened in the Original Poster's second example:
Executive summary [As Prof Lawler used to say]:
In yes/no questions we only move the auxiliary verb to the front of the sentence. The rest of the clause remains undisturbed. In other words the rest of the sentence is the same as it would be in a declarative clause.
In open questions we often have to move the interrogative phrase from another part of the clause. This can often break up larger constituents within the sentence, and also means that open questions can look considerably different from their declarative counterparts.
Authoritative source
All of the information above can be found in Rodney Huddleston, Geoffrey K. Pullum and Brett Reynolds (2022) A student's introduction to English grammar. Cambridge University Press.
Before going into the difference between do you know and do you think, we need to first think about the difference between open and closed questions. An open question has any number of possible answers, whereas a closed question has a closed set of answers (Yes and No).
Here are your examples:
(1) What time is it? [open question]
(2) Do you know what time it is? [closed question]
(3) What time do you think it is? [open question]
As shown, (1) and (3) are open questions, whereas (2) is a closed question. In (1) and (3), the speaker is asking the listener about the time, so the question starts with an interrogative phrase such as what time. In (2), on the other hand, what the speaker is asking the listener about is not the time but whether the listener knows something or not, so the question starts not with an interrogative phrase but with an auxiliary such as do.
Now, let's look at this pair:
(2') ?What time do you know it is? [open question]
(3') ?Do you think what time it is? [closed question]
These are strange things to say, not because of grammar but because of their meanings. In (2'), the speaker is asking the listener about the time that the speaker assumes the listener already knows, so adding do you know would be stating the obvious to the point where it sounds strange.
In (3'), the speaker is asking the listener about whether or not the listener thinks what time it is, which would sound strange in most contexts.
