Preheat oven to 300°F. Finely chop vegetables. In a medium skillet over medium heat, sauté vegetables and garlic in olive oil for about two minutes. Add salt and pepper. Transfer skillet to oven to roast vegetables, about five minutes. Return skillet to stove over medium-high heat. Add Chianti, bring to a boil, then reduce to a simmer. Cook until wine reduces to desired consistency. Taste and adjust salt and pepper as needed. In a blender, carefully purée vegetables, wine reduction, and balsamic vinegar to desired consistency (purée should be thick, not liquid).
Raise the oven temperature to 350°F. Rub bottom of iron skillet lightly with olive oil. Season lamb chops with olive oil, salt, and pepper. On stove, heat an iron skillet over high heat. Sear lamb chops for one minute per side. Transfer seared lamb chops to preheated oven to finish cooking, 15 minutes for medium rare.
Spoon puréed vegetables onto plate and arrange chops on top. Serve immediately.