Preheat oven to 440°F (225°C), with the rack placed as low as possible. Lightly grease a baking sheet with oil, set aside.
Split the dough into two balls. To make the first pizza, stretch and roll one piece of dough into an oval, with the edge slightly thicker than the middle. Transfer to the baking sheet. Brush it with one tablespoon of oil.
Take 1/2 of the mozzarella and 1/2 of cheddar and sprinkle them over the dough. Shape some of the cheese into little nests, where you want the eggs to go. (The cheese walls will hold the eggs in place.)
Crack each egg into a small bowl first, then slide it into the cheese nest. You want 2 eggs per pizza. Lastly, add 1/2 of tomatoes and onions.
Bake pizza in the bottom of the oven for 7 minutes. Then transfer the rack to the middle of the oven and keep baking, just until the egg whites are cooked and cheese bubbly, about 5-7 minutes. Remove from oven, sprinkle with parsley and serve.
Make the second pizza as the first one bakes and as you take one out, you place another one into the oven.