Melt 3 tablespoons butter in a microwave-safe bowl.
Stir in 3 tablespoons sugar, 2 tablespoons brown sugar, and 1 tablespoon cinnamon. Mix until smooth. Set aside.
Whisk together 2 eggs and 1/4 cup milk in a shallow bowl. Set aside.
In a small bowl, mix 2 tablespoons sugar and 2 teaspoons cinnamon. Set aside.
Trim crusts from each slice of bread.
Roll each slice to 1/8-inch thickness with a rolling pin.
Spread the butter mixture on each slice and roll tightly.
Repeat with the remaining bread and butter mixture.
Heat 3 tablespoons butter in a large non-stick pan over medium heat.
If using a small pan, use 2 tablespoons butter and cook in batches.
Dip each roll in the egg mixture and place in the pan.
Cook for 1-2 minutes per side until golden and crispy.
Remove from the pan and sprinkle with cinnamon-sugar mixture.
Serve immediately with maple syrup.