Preheat your oven to 425 degrees
Add 1 tablespoon of extra virgin olive oil into a large skillet over medium heat
Add the thinly sliced yellow onions and sprinkle with 2 teaspoons of granulated sugar
Cook over medium heat until the onions begin to soften and develop a lightly golden brown color
Sprinkle the onions with a big pinch of salt and black pepper and add a splash of stock
Cook until the stock is absorbed, stirring occasionally, and repeat the process, adding another splash of stock every few minutes until the onions are caramelized, soft and golden brown
While the onions are caramelizing, brown the Italian sausage
Once the onions and Italian sausage are cooked, set them aside for now
Make the béchamel by melting 2 tablespoons of butter in a small saucepan, whisking in 2 tablespoons of flour and cooking the mixture for about 30 seconds, stirring constantly
Slowly add 1/2 cup of whole milk while continuing to whisk, add a pinch of salt and black pepper and as soon as the béchamel thickens, remove from the heat
To assemble the pizza, unroll the pizza dough onto a pizza pan or wide flat baking sheet
Spread the béchamel thinly over the pizza dough leaving a 1/2” boarder around the outer edge
Top the béchamel with 4 ounces of finely shredded mozzarella
Add the caramelized onions and browned Italian sausage
Bake the pizza for 8 minutes and then transfer it back onto a heat proof surface leaving the oven set to 425 degrees
Use a large spoon to form four nests in the toppings and crack the eggs into those nests
Season the eggs with a pinch of salt and black pepper and sprinkle the remaining mozzarella over the pizza
Bake the pizza for 8-12 minutes at 425 degrees depending on how you prefer your eggs to be set
Once the pizza is baked, transfer it onto a heat proof surface
Top with fresh arugula and some crushed red pepper flakes (optional) and slice into wedges
Serve immediately with fresh fruit