Using a paring knife, make several small slits over the entire surface of the lamb, and stuff the slits with the sliced garlic. Add the yogurt, mint, cumin and hot pepper sauce to the work bowl of a food processor and process until smooth. Place the lamb on a large baking sheet and rub the entire leg with the yogurt mixture. Cover and refrigerate for at least 8 hours, or overnight.

Preheat the grill to high. Remove the lamb from the marinade and season well with salt and pepper. Place the lamb, skin side down, on the grill. Grill until the skin side is golden brown, then turn the lamb over, and reduce the heat of the grill to medium so that the grill maintains a constant temperature of 350 degrees F. Continue grilling until a thermometer inserted deep into the meat reaches a temperature of 145 degrees F for medium-rare doneness, about 1 1/4 to 1 1/2 hours. Remove lamb from grill, cover with foil, and let rest for 10 minutes. Slice the meat on the bias into 1/4-inch thick slices.