In a food processor, blend the basil, garlic cloves, pine nuts and olive oil until all ingredients are well combined but you still have quite a chunky texture. Add the pecorino Romano and stir through. Start with 1/4 cup and if you need more cheese slowly add in.
On a low heat in a large fry pan, heat 2 - 3 tablespoons of pesto in a pan. When the pesto is slightly sizzling, crack the eggs in and leave to cook on a low heat. Using a spoon, scoop up the hot oil from the pesto and drizzle over the egg whites until they are puffed, bubbly and cooked. Make sure the oil does not touch the egg yolk or it will cook it. It should take about 4 - 5 minutes.
Season the egg yolks with a pinch of salt and enjoy on some sourdough. Store the remainder of the pesto in an airtight jar for up to 2 weeks in the fridge.