Preheat oven to 425 degrees. Oil a 10×15-inch rimmed baking sheet.
Work your pizza dough around the baking sheet by pressing gently with your fingers and hands. It may take a little time to work the pizza dough out as it tends to pull back from the sides. Once it seems to stay put, use your hands to tuck and push a little of the dough up against the side of the baking sheet creating a crust. Bake in the oven for 7-8 minutes.
Beat the eggs and water together. Melt butter in a large nonstick skillet over medium heat. Add egg mixture and cook slowly, rotating them gently with a spatula. Be sure to slightly undercook the eggs as they are cooked again in the oven. Season the eggs with kosher salt and fresh ground black pepper Spoon the eggs onto the pizza dough.
Top the eggs with both kinds of cheese, cooked bacon, cooked sausage, red pepper, and scallions.
Place back in the oven for 10-12 minutes, turning on the broiler for the last 1-2 minutes. Rotate the baking sheet several times during the broiling process and stay close by, as I have always found broilers somewhat unpredictable.