Make broth by combining the lamb trimmings (or use lamb shank cut up), wine, garlic, mint and stock. Bring to a boil then reduce to a simmer for 1 hour. Skim occasionally.
Strain broth through a fine chinoise and set aside, remove fat from top as it cools and comes to the surface. (This will be more broth than you need for 4 chops)
In a hot saute pan with a little oil, saute lamb chops on all sides until medium-rare. Finish in the oven, if necessary. Set chops aside to rest for a few minutes while you make the sauce.
Wipe out the saute pan and pour in the red wine vinegar, reduce over high heat until almost dry. Add 1 cup of the lamb broth and continue to reduce by about 1/2. Reduce heat to low and swirl in the butter to lightly thicken the sauce.
Season with salt and pepper and a tiny pinch of sugar. If desired add more vinegar.
Stir in the chopped mint and any juices from the lamb chops.