Place a 6-in. cake pan or small ovenproof saucepan in the center of a 10-in. cast-iron skillet. Cut each roll into thirds; roll each piece into a ball. Place along outer edge of skillet. Gently stack remaining balls on top of bottom layer, leaving some space between them. Cover and let rise until almost doubled, about 30 minutes.
Preheat oven to 400°. Microwave butter, garlic powder and red pepper flakes, covered, until butter is melted. Brush half of butter mixture over dough. Reserve remaining butter mixture.
Bake until dough balls are set and beginning to brown, 15-18 minutes. Remove cake pan from skillet. Meanwhile, in a small skillet, heat butter over medium-high heat. Add mushrooms; cook and stir until tender, 5-7 minutes. Add garlic; cook 1 minute longer.
In a large bowl, combine the mayonnaise, cream cheese, 3/4 cup mozzarella cheese and 1/2 cup Parmesan cheese. Add the mushroom mixture, artichokes and spinach.
Spoon dip into center of skillet; sprinkle with red pepper and remaining 1/4 cup mozzarella. Brush rolls with remaining butter mixture; sprinkle with remaining 2 tablespoons Parmesan. Bake until dip is heated through and rolls are golden brown, 10-15 minutes. If desired, sprinkle with additional red pepper flakes.