Recently, Webster's university launched a Minor degree in Chess "Chess in Education"
(see this link).
There are a few educational institutions that offer courses on chess: TTU (under SPICE program mentioned in xaisoft's answer), UT Dallas, etc.
There is a SPICE office at Webster's university (see this link).
It seems that the idea of Aberdeen University to offer PhD in chess did not take off (no news after 2001; currently, no such program is offered in the university).
Discussion on degree in chess
I would like point out that an academic education program on chess could be for players, coaches or researchers (could there be one for chess authors who write books for players? I am not sure).
For players, performance in tournaments is indeed a better measure. But, an academic degree on chess could attract players who are not top performers but are very much interested in chess. This could help popularise chess. Also, teaching students how to think logically is an important function of education (at least science education). If this is the angle of approach, it makes a lot of sense to not limit to only chess and rather make it a degree on (logical and/or strategy) games. An example of such a course is MS in games for learning in NYU Steinhardt (with focus on use of games for teaching and learning).
In the current scenario, to get officially recognised as a coach (by fide or platforms such as chess.com or lichess.org), you need a high rating (2300+). This is reasonable if the coach is supposed to train class B or higher level players. But, for beginner coaches, this is not reasonable. Schools and universities would benefit from a way to recognise good coaches for beginners who are not necessarily themselves top rated players. Moreover, an expert may sometimes not know the typical issue of a complete beginner.
A quote from "Ten ways to know when a chess coach is good"
The master may be used to dealing with abstract theory and concepts but the new player is still struggling to lay down a solid foundation to build upon. Imagine how ineffective it would be, for both parties involved, if a mathematician at NASA were to teach 4 year olds how to count.
Hence, sports/game science degree on chess would be a valuable tool to recognise good coaches or beginners.
One could do a PhD in chess from the sports science angle (see the topics listed in lodebari's answer).
Apart from that angle also, research on sports and games is a genuine one (for instance, see the papers on chess listed on google scholar). Two papers on chess I find important are "On numbers and endgames: Combinatorial game theory in chess endgames" by Elkies, and "A Combinatorial game theoretic analysis of chess endgames" by Wu et al.
I personally think that combinatorial game theory has produced results on chess beyond the expectation, but thas seems not enough to garner serious attention yet (the sucess of ML/AI chess engines seriously reduced the chances of CGT-based chess reseach getting public attention). Instead of talking about combinatorial game theory here, I shall provide a link to my answer to the question What projects have been undertaken to solve chess completely? (also see this answer to another question).