Can an omnipotent being do something that is illogical, yet logical at the same time?
No, but that's not a limitation of omnipotent beings. That's a limitation of sentences. A sentence has a subject and a predicate, which modifies the subject. For example, take the sentence, "The cat is on the mat." The subject is "cat" and the predicate, "is on the mat". The predicate tells us something about the subject, in this case, where the cat is, on the mat.
So, for a sentence to make sense, it needs a valid subject and a valid predicate. What I mean by valid is, they have to refer to something. A sentence like "The cat is prep gwarlek." has a predicate, "is prep gwarlek" but unlike the previous predicate "is on the mat", this new one, "is prep gwarlek", fails to refer to anything. Another way in which a string of words can fail to refer to anything is by saying something absurd like "four-sided triangle".
The way in which "four-sided triangle" fails to refer to anything is a little different from how "prep gwarlek" fails. The non-referring phrase "prep gwarlek" is built out of meaningful letters but the individual words fail to refer, which of course causes the phrase as a whole to fail to refer. However, the non-referring phrase "four-sided triangle" is built out of both meaningful letters and meaningful words but the phrase as whole fails to refer. The reason why is as follows.
When we build up phrases like "brown three-legged male dogs" we are referring to the members that are common in all of these three sets, "brown dogs", "three-legged dogs" and "male dogs". We want to refer only to members that are in all three sets instead of just one or two. That's called an intersection of sets.
In the same way, "four-sided triangles" refers to the intersection of the set of "four-sided things" and "three-sided things". But since these sets are disjoint, having no members in common between them, the intersection refers to nothing at all.
So, the problem with asserting that four-sided triangles can't exist is that you aren't referring to anything. Also, the problem with saying that God can do something that's both logical and illogical is that you aren't referring to anything. The set of logical things and the set of illogical things are disjoint. They have no members in common. Referring to their intersection is referring to nothing at all.