You can get virtually any masters degree in ~ 2 years if you put the effort into it. The question is why do one? You should be spending the time doing something that you enjoy and ideally it would be nice if it advances your career. When you get a good idea of what you want your career to look like, it will become obvious what can help your career. This can range from basic science bench work to translational work (e.g. clinical trials work) to purely clinical outcomes work to policy work.
Things I have seen surgery residents do with 2 years of research time:
Degrees:
MBA
MPH
MPP (Masters in Public Policy)
Masters of Divinity
Masters in Clinical Trial Design (or something like it)
MHA or MHSA (Masters in Health Science Administration)
Fellowships open to people from all institutions:
NIH (Many research programs here, NCI probably recruits surgeons most heavily)
Robert Wood Johnson Foundation Clinical Scholars Program (based at UCLA, U-Michigan, Yale, Penn)
Critical Care Fellowship (Many fellowship allow you to start after your 3rd clinical year in surgery)
Surgical Innovation fellowship (based at Stanford)
Many other research positions are also available at almost every well-funded institution.
PhD (often requires 3 or more years in research)
I'm sure there are many more. Those are just off the top of my head.
As far as how the degree programs work, typically you need to apply the year before you wish to enter. From MPH and MHAs typically most public health schools will accept your MCAT, but some will require the GRE (most common) or GMAT (less common). Most will give you some credit for your med school courses, typically shortening your 4 semesters down to about 3 on average.
MBA institutions typically require the GMAT, although some institutions with programs targeted to doctors will waive that requirement or use your MCAT (rare). Again, you should be applying the year prior. Some cross-credit is often allowed, again allowing you to shave off about 1 semester if you wish.
Don't really know much about the JD programs, but you'd be looking to take the LSAT, and it typically takes 3 yrs, but not sure if they would give you any cross credit. Seems unlikely, but possible given that MD/JD programs typically run for 6 and not the 7 that it would take to do both separately.