How to Make It
Preheat oven to 400°F. Stir together shredded mozzarella, pepperoni, oregano, and 1/2 cup of the sauce in a medium bowl.
Form 8 thin patties of beef. Divide pepperoni mixture over 4 of the patties; top each with a second patty, and firmly press edges together to seal in filling. Sprinkle patties with salt and pepper.
Heat a large cast-iron skillet over medium-high. Cook patties 4 minutes on each side. Top each burger evenly with sautéed mushrooms and sliced mozzarella, and place skillet in preheated oven. Bake until cheese is melted, about 2 minutes.
Combine olive oil and garlic in a small bowl; brush oil mixture on bun halves. Bake at 400°F until lightly toasted. Spread remaining 1/4 cup pizza sauce on each of the French bun halves. Place a burger on bottom bun; top each with basil and top of bun.
Thread 2 olives and 1 pepperoncini onto each of 4 (6-inch) skewers, and stick 1 skewer into each burger.