What can we change?

First off, let me say something which will keep you out of hot water: different people have different learning styles and this is not necessarily any sort of "mental condition" -- it could be, but it might not be. Therefore tact requires that you completely stop being suspicious of the latter: if you are sneakily telling yourself "it's okay, she's just autistic" then you strongly risk accidentally blurting that out to either a friend or a colleague, and once that's out of the bag your job could be impacted by the results. Just admit honestly to yourself that you don't know, and that it is one of the possible explanations for why your student is a "problem student".

Now, I agree that the student's learning style shares some similarities with some aspects of Asperger syndrome or possibly something on the autistic spectrum, regardless of whether or not he/she has those. So, what changes can you make that would be friendly to that sort of learning style?

Use more neutral language.

If you are saying "Here is how you solve that problem" then you might be silently implying that either (a) there is only one way to solve it, or (b) your way is the best way. There's usually a lot of different ways to skin a cat.

For instance, in my field of physics, very often there is some thermodynamic property which you can easily see by looking at fuzzy things called "total differentials" but which would have to be more rigorously made by appealing to certain definitions of certain limits -- someone who has done all of the needed rigorous work will probably come to the easy way and say "yes, but that's a load of crap, you do not know that these properties hold!" and indeed they're right, it's a heuristic rule (not a proof) to re-derive the property.

Similarly in computer science, e.g. we do amortized analysis by appealing to "potential functions" which we choose to make our proofs work; someone who has actually carefully analyzed the algorithm and the time costs of every little part will probably think that your approach is full of crap. Opt instead for vocabulary like "Here's one way to see that this is the right answer..." that makes it more clear that you're not telling people how it has to be done, but just how you like to do it.

Make the class a little more predictable.

This learning style benefits from being able to review material in advance, on their own. Don't be afraid to ask them to review the material in advance to understand how it all works.

Make the class a little more structured.

Right now, you have advertised that you're open to all questions at all times, and you're getting upset with a student for asking all of his questions whenever he has one! I get that you're also upset with his tone and whatever, but you need to change that tendency.

To fit with the last point: always introduce this by saying "hey, at our next class, as a heads up, I am going to slightly change the rules of this recitation section...". Remember, if you're making a change it has to be predictable!

Try to either make a slideshow or write on a board so that you never have to erase anything; if there are multiple slides/boards, number them. This gives a consistent way to reference what's already been said and when, and removes the pressure to get in a question before the board is erased.

Have clear rules about how questions will be asked and when they will be answered. For example, tell everyone that whenever someone has a question, you'll judge whether you can answer it in less than 30 seconds or not; if not then you'll write it down in a notebook with the slide number and the asker, and then you'll keep going -- 15 minutes before the end of class you'll have an alarm go off to remind you to go into "questions mode" where you go back and answer these questions in sequence. If this does not reduce the number of questions, again, give advance notice that each student will receive a maximum of 5 questions to ask during that period. The important thing is that this remains consistent for everyone -- if any question takes longer than 30 seconds to answer, write it down, come to it at the end of class. They must be firm rules: "If you have more than 5 questions that I have to write down, I'll take photos of these boards with my phone and will answer them during my office hours."

If you can get a 30 second hourglass that's even better, because then you can advertise it to everyone else as a sort of game: tell people "I'm trying to get better about quickly and concisely getting to the heart of the problem!" and people will respond well.

Try to set up an incentive system.

For example, maybe you will always try to resolve every question in 30 seconds with your little hourglass timer, but you will up-front either say that a question is "easy" or "hard" for you. Now buy a big bag of candy or some other sort of reward (stickers are still fun, right? Or whatever). Then the rule can be: if someone says that their "hard" question was resolved after 30 seconds of explanation, you both get a candy. This encourages the student to still ask their question, but to maybe try to "boil it down" to some essential basis that can be answered much more quickly.

What do the above approaches center around?

Those sorts of changes encourage that the "problem student" gets to study your approaches on his/her own, chooses 5 questions that really speak to their own conceptual difficulties with your approach, and has a set time where they get their questions answered. In addition they will hopefully try to reduce the scope of their problem to something precisely wrong with what your approach is, rather than trying to advocate their approach in place of yours.

This student's learning style is actually really refreshing! They are not a robot -- but they do really just want things to be simple and consistent. To understand the human side, think of it as a "cut through the BS" type of learning style: all of these extra rules and social considerations are a huge extra mental cost where you're trying to negotiate certain social relationships as well as get your questions answered to your satisfaction -- but why accept that sort of extra mental load if you don't want to?

Rituals and rules help to structure that social-relationship-negotiation and make it cognitively easier. Yes, it is added administrative cost up front, but it can also pay off in giving you more freedom with your limited resources later. This learning style is refreshing precisely because it challenges you to essentially always "do the dishes right after you eat" -- it sucks because you have to interrupt your TV show or whatever to clean up, but it turns out that it is more efficient (the dishes are much easier to clean before the food dries onto them) and has a certain aesthetic benefit (your apartment is cleaner on average). So take this as a time to try and learn from them, and to pick up skills that will better help you teach people in general in the future.