1. Heat 3 1/2 tablespoons of the oil in large skillet set over medium-high heat. Add onions and sugar; sauté, stirring often, until softened, 7 to 8 minutes. Reduce heat to medium-low. Continue to cook, stirring often, until onions are sweet and tender, 7 to 8 minutes longer. Stir in mint and vinegar and season to taste with salt and pepper. Set onion mixture aside. (Can be covered and refrigerated up to 1 week.)
2. Heat broiler. Place bread on large baking sheet; broil bread until golden brown on both sides.
3. Remove baking sheet from oven. Rub garlic over toast tops. Brush remaining 2 1/2 tablespoons oil over bread. Divide onion mixture among slices, then sprinkle with cheese.
4. Broil until cheese just melts. Transfer bruschetta to large platter and serve immediately.