In a small microwave-safe bowl, melt 3 tablespoons butter until just melted. With a fork, stir in 3 tablespoons granulated sugar, 2 tablespoons brown sugar, and 1 tablespoon cinnamon until a smooth spreadable mixture forms. Set aside.
In a shallow bowl, whisk together the eggs and milk. Set aside.
In another small bowl, stir together 2 tablespoons granulated sugar with 2 teaspoons cinnamon. Set aside.
After trimming the crusts from each slice of bread, roll one slice out to about 1/8-inch thickness with a rolling pin. Spread each slice with a thin layer of the butter mixture and roll up tightly. Repeat with remaining bread slices and butter mixture.

Place 3 tablespoons butter in a large non-stick sauté pan and let melt over medium heat. (If using a small sauté pan, use just 2 tablespoons butter and sauté the roll-ups in two batches.) Dip one roll at a time in the egg mixture and place in the sauté pan. Allow roll-ups to sauté for 1-2 minutes per side until golden brown and slightly crisp on the outside.
Remove to a work surface and sprinkle each roll-up generously with the cinnamon-sugar mixture.
Serve immediately with maple syrup.