Start by preheating the oven to 350 degrees. Grab a 9X13 inch baking pan and coat it with nonstick spray.
Take the entire package of rolls out of the package, but DO NOT separate them. Put the bottom halves of the rolls into the baking pan, still together in one piece.
Then, take half of the cheese slices and put them on the rolls. Followed by the ham layers, the turkey layers, and then add the remaining half of the cheese slices right on top.
After the meat and cheese are stacked, put the top halves of the rolls right on top.
Grab a small bowl and whisk together the egg, pepper, and melted butter. Use a pastry brush and brush this wet liquid over the top of all the rolls.
Put the dish in the oven and bake for 25 minutes, or until the top of the buns are golden brown and the cheese is melted. If you happen to notice that the buns are looking too brown, you can make a tent tin foil over the top of the buns loosely. (Just make sure to tent it so that it doesn't stick to the very top of the buns!)
Once the buns are done baking in the oven, remove and sprinkle on a little bit of powdered sugar and serve with some jam or maple syrup. (optional)