Whisk balsamic vinegar, olive oil, and soy sauce together in a bowl; stir in chopped mint. Add lamb chops and toss to evenly coat. Cover with plastic wrap and marinate in the refrigerator for 8 hours to overnight.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Place lamb chops on the grill, reserving marinade in the bowl. Cook lamb chops, flipping once, until slightly pink in the center, about 5 minutes per side. An instant-read thermometer inserted into the center should read at least 145 degrees F (63 degrees C). Place onto a serving plate.
Pour reserved marinade into a small saucepan; bring to a boil. Reduce heat to low. Simmer until marinade is reduced by half, about 10 minutes. Drizzle over lamb chops; garnish with mint leaves.