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 ⅛-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 saute pan and let melt over medium heat. (If using a small saute pan, use just 2 tablespoons butter and saute the roll-ups in two batches) Dip one roll at a time in the egg mixture and place in the saute pan. Allow roll ups to saute 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.