Combine tomatoes, bell pepper, onion, balsamic vinegar, 1/4 cup olive oil, basil, and garlic; let tomato mixture rest for 15 to 30 minutes.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Arrange French bread slices on a large baking sheet; drizzle with 1/4 cup olive oil.
Toast bread in preheated oven until lightly browned, about 1 minute on each side. Top bread slices with tomato mixture using a slotted spoon, allowing excess liquid to drain. Return bread to baking sheet and sprinkle with mozzarella cheese.
Broil in preheated oven until cheese melts, about 3 minutes. Serve immediately.
