1. Adjust oven rack to about 4 inches from heating element and heat broiler; broil bread slices until golden brown on both sides. Brush both sides of each slice with oil and rub with garlic clove.

2. Heat 3 tablespoons oil, bell peppers, onions, and ½ teaspoon salt in 12-inch skillet over medium-high heat; cook, stirring occasionally, until vegetables are softened and browned about the edges, 10 to 12 minutes. Reduce heat to medium, push vegetables to side of skillet, making a clearing in center, and add remaining 1 teaspoon oil, garlic, and red pepper flakes to clearing; cook, mashing garlic with wooden spoon, until fragrant, about 30 seconds, then stir into vegetables. Reduce heat to low and stir in tomatoes, reserved juice, and thyme. Cover and cook, stirring occasionally, until moisture has evaporated, 15 to 18 minutes. Off heat, stir in vinegar and remaining ¼ teaspoon salt.

3. Divide pepper mixture evenly among bread slices, top with shaved Parmesan, and serve.