Preheat oven to 350°F. Lightly butter the bottom and sides of a rimmed quarter sheet pan (or a 9 x 13-inch baking dish) and set aside.
In a large bowl, whisk the eggs along with salt and pepper. Pour eggs into the sheet pan and carefully transfer pan to oven. Bake for about 12-15 minutes, until eggs are just set and no longer wobble when you jiggle the pan.
While the eggs are cooking, heat a large skillet over medium heat. Lightly butter the insides of the rolls and place on the skillet, butter-side down. Griddle until golden brown, rotating rolls as necessary until all are griddled.
Whisk together the mayonnaise and as much hot sauce as you’d like in a small bowl. Spread a bit on both sides of all the rolls.
When the eggs are done, slice into 6 pieces and add one to the bottom of each roll then immediately top with a piece of cheese and two slices of bacon. Add the top bun, then wrap the sandwiches up in parchment or foil to melt the cheese and steam the rolls a bit.
Slice in half and enjoy.