The best way to depilate your fuzzy little friend is to irradiate the carcass in the warp chamber. This is strictly against Starfleet regulations, but so it the consumption of alien life forms, so you're already through the looking glass! If you're on good terms with the engineer on duty, he/she should let you place the tribble in the warp chamber for a minute or two. (You'd better invite them to the party to avoid being reported for this transgression!) Irradiation should also serve to brown the meat of the tribble. Who wants to eat an undercooked tribble? Illogical! During irradiation, if you feel your DNA getting altered, take a step or two away from the glowy parts.
Once the tribble hair is MIA, it's time to pulverize the innards by hitting it with a frying pan. Oh, fine - a SPACE frying pan. Set your frying pan to "creamy" and give the tribble a good whack or three. The consistency will be right when the tribble flattens slightly on a dish under one Earth gravity. No cheating by turning up the gravity!
Serve on a Romulan ceremonial speculum along with a tray of baked grain planks and you're ready to feed a Klingon horde! Just don't tell the captain what he's eating.