Combine carrots and 1/4 cup water in a large lidded skillet over medium-high heat. Cover and cook 4 minutes.
Meanwhile, remove 16 leaves from lettuce, rinse and pat dry. Set aside.
Stir canola oil into skillet with carrots. Add mushrooms, sweet pepper and garlic. Cook 2 minutes. Crumble in ground beef and cook 3 minutes. Add brown rice and cook 2 minutes. Lightly beat egg; push skillet contents to one side of pan. Add egg to skillet and scramble, 1 minute. Stir in scallions, soy sauce, vinegar, sugar, sesame oil and ginger. Remove from heat.
Serve meat mixture with lettuce leaves (spoon a scant 1/3 cup into each leaf). Sprinkle with mint leaves and serve with steamed broccoli on the side.