Make Butter-Cinnamon Filling: In a small microwave-safe bowl, melt 3 tablespoons butter until just melted. Stir in 3 tablespoons granulated sugar, 2 tablespoons brown sugar, and 1 tablespoon cinnamon with a fork until smooth and spreadable. Set aside.
Prepare Egg Mixture and Coating: In a shallow bowl, whisk together eggs and milk. In another small bowl, stir together 2 tablespoons granulated sugar with 2 teaspoons cinnamon. Pour onto a plate for coating. Set both aside.
Prepare Bread: Trim crusts from each bread slice. Using a rolling pin, roll one slice to about 1/8-inch thickness. Spread with a thin layer of butter-cinnamon mixture. Starting from one end, roll up tightly. Repeat with remaining bread slices.
Cook Roll-Ups: Place 3 tablespoons butter in a large non-stick sauté pan over medium heat and let melt. (Use 2 batches if pan is small.) Dip one roll at a time in egg mixture, coating all sides, and place seam-side down in pan. Cook 1-2 minutes per side until golden brown and slightly crisp, turning to brown all surfaces evenly.
Coat and Serve: Remove cooked roll-ups to work surface and immediately roll in cinnamon-sugar mixture while hot, coating all sides generously. Serve immediately with warm maple syrup for dipping.