Prepare. Preheat your oven to 350 degrees Fahrenheit with your pizza stone inside. Line a baking sheet with parchment paper. Thinly slice and thoroughly wash the leek. Cut the potato into ⅛” to ¼” slices. Crack the eggs into individual bowls and set aside (do not put both in one bowl).
Roast the potato. Gently toss the potato slices in a bowl along with enough olive oil to generously coat them, salt to taste, and a generous amount of black pepper. Arrange in a single layer on the prepared baking sheet and roast at 350 degrees Fahrenheit until cooked through and beginning to develop some brown edges, around 30 minutes.
Cook the leek. While the potatoes roast, melt the butter in a skillet. Add the leeks, a generous pinch of salt, and chili flakes (optional). Cook on medium-high, stirring frequently, until tender, 5 to 7 minutes.
Stretch and top the dough. Stretch your pizza dough to a 12” to 14” circle. Brush the entire top surface with olive oil, then top with the mozzarella cubes. Sprinkle on the leeks, then arrange the potato slices on top. As you do all of this, leave two empty little “nests” that you can pour the eggs into later.
Partially bake the pizza and add the eggs. Bake the pizza at 350 degrees Fahrenheit on your preheated pizza stone until the crust puffs up and just barely begins to change color. At this point, pour the eggs into the two “nests” that you left for them.
Finish the pizza, and enjoy! Return the pizza to the oven and continue baking until the egg whites are set, the yolks are runny, and the crust is golden brown. Finish the pizza by sprinkling the egg yolks with sea salt and cracking some extra black pepper over the whites, then shaving fresh parmesan on top of everything. Enjoy!