Puree the balsamic vinegar, garlic, and thyme in a blender until mixed. With the blender running, slowly pour in the olive oil until thickened and incorporated. Pour the marinade into a resealable plastic bag. Add the pork pieces, coat with the marinade, squeeze out excess air, and seal the bag. Marinate in the refrigerator for 30 minutes.
Preheat an outdoor grill for medium heat, and lightly oil the grate. Remove the pork from the marinade, and shake off excess. Discard the remaining marinade.
Cook the pork on the preheated grill until no longer pink in the center, about 10 minutes. An instant-read thermometer inserted into the center should read 145 degrees F (63 degrees C). Once cooked, remove from the grill, cover with aluminum foil, and allow to rest for 5 to 10 minutes before slicing.
While the pork is cooking, melt the butter in a saucepan over medium heat. Stir in the brown sugar until it begins to simmer. Add the apples, cherries, ginger, cinnamon, and nutmeg. Cook and stir until the apple is tender, about 5 minutes. Serve the sliced pork topped with the apple compote.