Crack the eggs individually into 2 very small bowls and let them sit on the countertop at room temperature for at least 30 minutes while you prep the remaining ingredients. NOTE these bowls should be small enough so that you will be able to accurately shimmy each egg into a premade 'well' or dip you will create in each dough.
Remove the pizza dough ball from its packaging and place on a lightly floured work surface to rest for 20 minutes while you make the basil oil.
To the barrel of a mini food processor or chef prep, add the loose basil leaves, olive oil, and a small garlic clove and process until smooth. Taste for the need for salt and pepper and add if necessary. Process again to combine. Transfer to a pretty serving bowl with spoon or a creamer.
Preheat oven to 450°F with rack in upper third position. Dust a baking sheet with cornmeal and set aside. Use a bench scraper or long sharp knife to cut the now rested dough in half to equal two portions. Flatten and press and roll and stretch one piece at a time to a large oblong about 1/4-inch thick and around the size of a small dinner plate and transfer to the prepared baking sheet.
Sprinkle doughs evenly with shredded smoked mozzarella, the finely planed pecorino, and some mushroom slices, leaving a small round-ish well in the middle of each and a 1/2-inch border for the crust. The idea behind the well is that this is where the egg will reside, so make the well wider than deep and abstract if round does not conform. 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 8 minutes.
Remove pizzas from oven. Using the back of a spoon, enhance the well in center of each pizza by gently pressing down. Take each egg and guide it into a dough well while shimmying the yolks to rest in the center. NOTE if your well is not wide enough, the whites may run over the sides of the pizza, and while this is okay, ideally, you want it to stay confined to the well.
Carefully return pizzas to oven and bake at 450°F until egg whites are set but yolks are still runny, 5 to 7 minutes. A good way to tell if your whites are set is to jiggle the baking sheet. If the whites are set, you should see little to no jiggling. NOTE if you plan to enjoy this as a handheld, consider the doneness of the yolk and bake until only slightly runny.
Use a large, wide metal spatula to remove each pizza to a plate. Serve with basil oil. Allow eaters the choice of a knife and fork or simply go in with both hands like I do!