Preheat the oven to 425°F / 220°C.
Brush a 10x15-inch rimmed baking sheet evenly with olive oil.
Place the pizza dough onto the prepared baking sheet.
Gently press the dough across the baking sheet using the fingers and palms of the hands.
Continue working the dough toward the edges of the pan. If the dough pulls back, allow it to rest briefly, then continue pressing.
Once the dough remains in place, press a small amount of dough up along the sides of the baking sheet to form a crust.
Bake the crust for 7 to 8 minutes, or until it is lightly set. Remove from the oven and set aside.
In a medium bowl, whisk together the eggs and cool water until fully combined.
Melt the butter in a large nonstick skillet over medium heat.
Add the egg mixture to the skillet.
Cook the eggs slowly, gently moving them with a spatula as they begin to set.
Remove the eggs from the heat while they are still slightly undercooked, as they will continue cooking in the oven.
Season the eggs with kosher salt and freshly ground black pepper.
Spoon the scrambled eggs evenly over the partially baked pizza crust.
Sprinkle the shredded provolone and Pepper Jack cheese evenly over the eggs.
Top with the chopped cooked bacon, browned breakfast sausage, finely diced red pepper, and chopped green onions.
Return the pizza to the oven.
Bake for 10 to 12 minutes, or until the cheese is melted, the crust is golden, and the toppings are heated through.
During the final 1 to 2 minutes, turn on the broiler if additional browning is desired.
Rotate the baking sheet as needed during broiling and monitor closely to prevent burning.
Remove the breakfast pizza from the oven.
Allow it to rest briefly before slicing into 8 pieces.
Serve warm with salsa, if desired.