Butter, for greasing
2 tablespoons extra-virgin olive oil
1/2 cup finely diced pepperoni (2 ounces)
1/2 pound shiitake mushrooms, stems discarded and caps cut into 3/4-inch pieces
1 medium onion, minced
1 red bell pepper, cut into 1/2-inch pieces
2 teaspoons kosher salt, plus more for seasoning
8 large eggs
3 cups whole milk
1 tablespoon Dijon mustard
1 tablespoon soy sauce
1/2 teaspoon pepper
3/4 pound day-old challah, sliced 1 inch thick and cut into 1-inch dice (10 cups)
6 ounces Black Forest ham, finely diced (1 1/4 cups)
1 cup shredded Monterey Jack cheese (1/4 pound )
1 cup shredded aged white cheddar cheese (1/4 pound )
1/2 cup finely chopped scallions, plus thinly sliced scallions for garnish
Hot sauce, for serving