What are the FAA regulations on operating UAVs in class E airspace above FL600?
The FAA limits drone operations to 400 Ft. so getting to FL600 is not really possible. 


  Cannot be flown higher than 400 feet above ground level (AGL), unless
  flown within a 400-foot radius of a structure and does not fly higher
  than 400 feet above the structure’s immediate uppermost limit;


The FAA does offer a waiver process for flying drones beyond the regular part 107 rules. You can find all the info on that here. The request to operate in controlled airspace form does ask for a max altitude so its possible it can be requested. 



However someone in Russia appears to have flown their drone up to FL330 so it may be allowed there.