Preheat the oven to 450˚ F. Line 2 baking sheets with parchment paper. Divide the pizza dough into 4 pieces and roll each into a ball. Gently stretch each ball of dough into an 8-inch round, leaving a thicker crust around the edge; transfer to the prepared baking sheets. Prick the center of each dough round with a fork. Bake until lightly golden, about 10 minutes.
Meanwhile, thinly slice the mozzarella and dice the ham.
Brush the crusts with 2 tablespoons pesto, leaving a 1/2-inch border; top with the mozzarella and ham. Crack 1 egg on top of each pizza and return to the oven until the crust is browned and the egg whites are set but the yolks are still runny, 12 to 14 minutes. Top with the remaining 3 tablespoons pesto and sprinkle with Parmesan.