Heat your oven to 450F. Grease a baking sheet with cooking spray.
Divide the dough into 4 equal pieces. Stretch each piece into a round disk that’s as thin as you can get it. Arrange the discs on the baking sheet.
Brush each piece of dough with olive oil and season with Italian seasoning. Scatter ⅔ of the cheese over the dough, leaving a ½-inch border around the edge of each pizza. Top the pizzas with butternut squash, mushrooms, bell pepper, and potatoes. Bake for 8 minutes, or until the crust begins to set around the edges.
Crack an egg into a small cup, then carefully pour it onto one pizza. Repeat with the remaining eggs. Return the pan to the oven and bake 8-10 minutes, or until the eggs are cooked to your liking.
Remove the pizzas from the oven and sprinkle with green onions. Season with salt and pepper.