Line 2 baking sheets with parchment. Heat 3 tablespoons of the olive oil in a large cast-iron skillet over medium-high heat; add the onions and cook until softened and slightly browned, about 20 minutes. Stir in the oregano, garlic and red and yellow peppers and cook for another 5 minutes. Set aside to cool for 20 minutes.
Heat the remaining 1 tablespoon olive oil in another large cast-iron skillet over medium-high heat. Add the sausage and cook, using the back of a wooden spoon to break up the sausage meat into small bits, until browned, 10 to 15 minutes. Set aside to cool for 20 minutes.
On a lightly floured surface, roll out each puff pastry sheet to about 13 inches long and 10 inches wide. Place one of the puff pastry pieces on each lined baking sheet; set aside the other two. Whisk the egg in a small bowl with 1 tablespoon water.
Sprinkle 2 cups of the Monterey Jack cheese over one of the puff pastry sheets on a baking sheet, leaving a 1/2-inch border around the edges of the pastry. Continue layering with half the caramelized onions and peppers, sausage and pepperoni; finish with half the mozzarella slices. Brush the border of the puff pastry with the egg wash, lay one of the remaining puff pastry pieces over top and use a fork to crimp the edges. Using a sharp paring knife, cut three 1 1/2-inch vents in the top. Put the pie into the fridge to chill for 30 minutes before cooking. Repeat with the other puff pastry pieces and layering ingredients.
Preheat the oven to 425 degrees F.
Brush one of the pies with the egg wash. Bake on the bottom rack until the pastry is nicely browned, about 20 minutes.
Wrap the second pie in plastic wrap on the baking sheet, then in aluminum foil; freeze. When ready to bake, preheat the oven to 425 degrees F. Unwrap the pie and brush it with egg wash. Bake from frozen on the bottom rack until the pastry is nicely browned, about 30 minutes.
The egg wash amount is for one of the pies. When heating the second pie, use an additional egg mixed with water for the egg wash.