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Follow the Best Pizza Dough recipe to prepare the dough. (This takes about 15 minutes to make and 45 minutes to rest.)
Place a pizza stone in the oven and preheat to 475°F. OR preheat your pizza oven (here’s the pizza oven we use).
Slice the green parts off of the scallions so they form long ribbons. In a small skillet, heat 1/2 tablespoon olive oil. Add the scallions and sauté until slightly tender, stirring, for 1 minute. Remove from the heat.
When the oven is ready, dust a pizza peel with cornmeal or semolina flour. (If you don’t have a pizza peel, you can use a rimless baking sheet or the back of a rimmed baking sheet. But a pizza peel is well worth the investment!) Stretch the dough into a circle; see How to Stretch Pizza Dough for instructions. Then gently place the dough onto the pizza peel.
Brush olive oil over the dough and sprinkle with kosher salt. Sprinkle with the mozzarella cheese, then the Pecorino cheese. Top with scallions. Carefully crack two eggs on top and sprinkle with a bit of kosher salt.
Transfer the pizza to the pizza stone using the pizza peel, and bake until the egg white is set and the yolk is still runny, about 7 minutes.