Shred all cheeses on the large holes of a box grater. Coarsely shred butter on the large holes of box grater. Toss cheeses, butter, shallot, garlic, parsley, and red pepper in a medium bowl; season generously with salt and pepper. Transfer one-third of cheese mixture to a small bowl; set aside. Add dough pieces, a few at a time, to remaining cheese mixture in original bowl and toss gently to coat.

Lightly coat a medium cast iron skillet with nonstick spray. Arrange dough in skillet in an even layer; pour any leftover cheese mixture in the bowl over the pizza dough balls. Let sit in a warm place until beginning to puff slightly, 30–40 minutes.

Meanwhile, preheat oven to 375°. Bake bread until top is lightly browned and cheese is melted, 18–22 minutes. Top bread with reserved cheese mixture. Continue to bake until edges of bread are golden brown and firm and cheese is melted and oozing, 10–15 minutes more. Serve hot with marinara sauce and basil vinaigrette.