Place an oven rack 6 inches from heat source and preheat broiler to high. Warm 2 Tbsp. butter in a medium skillet over medium heat. Add onions and bell pepper; season with salt and pepper. Cook, stirring occasionally, until veggies are tender and golden, about 12 minutes.
Place bread slices on a baking sheet and lightly toast under broiler for 1½ minutes per side, checking frequently to avoid burning. Remove toast from oven, and place 1/4 of Brie slices on each piece. Broil until cheese is warm and has softened, 3 to 4 minutes. Remove each piece to a plate; divide onion-pepper mixture evenly among servings.
Melt remaining 1 Tbsp. butter in a large nonstick skillet over medium-high heat. Crack eggs into skillet and cook until whites are set but yolks are still runny, 2 to 3 minutes. Carefully place a fried egg on top of each piece of toast. Sprinkle with salt and pepper and serve immediately.