Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Spread 1 tablespoon butter over bread slices. Place buttered side-down in a grill pan; cook until crispy, about 3 minutes. Place 2 bread slices grilled side-down on a baking sheet.
Combine the remaining 3 tablespoons butter, parsley, Parmesan cheese, mustard, garlic, and pepper in a bowl. Spread over bread on the baking sheet.
Broil in the preheated oven until golden brown, about 4 minutes.
Place bacon in a large skillet and cook over medium-high heat, turning occasionally, until evenly browned, about 10 minutes. Drain bacon slices on paper towels.
Place eggs in the same skillet; cook over medium-high heat until whites are set on the bottom, 2 to 3 minutes. Flip over to cook the other side until whites are completely set, about 1 minute more. Layer bacon and eggs over the broiled bread and top with the 2 grilled slices.