Can a med student ask for his professor to teach a specific procedure?

Not really.

You can ask about a specific procedure and how it's done, and if it's a routine procedure that is part of the curriculum, or within a student's capabilities then they might well teach you to do it. However, you would not be able to do a more in-depth procedure, especially one outside your residency. Something you needed to be a resident to even attempt you would also not be trusted with to do on your own anyway.

Sometimes particular procedures are taught to doctors outside usual residency, but even then this is usually advanced stuff well beyond a med student, and in most cases it's for subspecialties and the like.

So for instance, if you wanted to learn a bit more about, say injections, then sure, they might teach you about that. They would never teach you how to operate on someone, though, unless you were a surgical resident.

There are a few different techniques for crutiate repairs in animals. Some procedures such as the TPLO is taught during residency and usually only done by a board certified surgeon. Typically larger breed dogs need this procedure (but this a point of contention among surgeons).

Private practitioners often perform a different technique called a lateral fabellar suture technique. It has the advantage of being cheaper, easier to learn. This procedure is most appropriate for smaller breed dogs. There is a lot of debate in the veterinary community about which procedure is "better". But, many veterinary clinics have someone on staff willing to perform this surgery.

You can ask anything you want. That doesn't me they will comply.