Step 1

Process basil, oil, and garlic in a food processor until smooth, about 25 seconds. Transfer to a small bowl; season with salt and black pepper to taste.

Step 2

Preheat oven to 450°F with racks in upper third and lower third positions. Lightly grease 2 baking sheets with oil. Dust 1 dough piece with flour, and place on a lightly floured work surface; stretch into a 7-inch circle. Transfer to prepared pan. Repeat with remaining dough pieces.

Step 3

Sprinkle dough circles evenly with mozzarella, pecorino, and mushrooms, leaving a 1/2-inch border for the crust. Sprinkle pizzas with salt, black pepper, and crushed red pepper to taste. Bake in preheated oven until crusts are puffed and starting to brown, 7 to 9 minutes.

Step 4

Remove pizzas from oven. Using the back of a spoon, make a small well in center of each pizza. Crack 1 egg into each well. Return pizzas to oven, rotating pans between top and bottom racks from previous bake. Bake at 450°F until egg whites are set but yolks are still runny, 5 to 7 minutes. Serve with basil oil.