Step 1

Heat the oven. Arrange a rack in the middle of the oven and heat to 375°F. Coat a baking sheet with cooking spray.

Step 2

Make the cheese sauce. Melt the butter in a small saucepan over medium-high heat. Add the flour and cook, stirring, until the butter-flour mixture loses its sheen, about 1 minute. Whisk in 1 cup of the milk and bring to a boil, whisking occasionally, until thickened, 3 to 4 minutes. Remove from the heat and whisk in 1/2 cup of the cheddar cheese and 1/4 teaspoon of the salt. Set aside to cool slightly while you make the rest of the toppings.

Step 3

Cook the sausage and eggs. Heat a medium nonstick frying pan over medium-high heat. Crumble the sausage into the pan and cook until browned and cooked through, 5 to 7 minutes. While the sausage cooks, whisk together the eggs, remaining 1/4 cup of milk, and remaining 1/4 teaspoon salt in a medium bowl. Use a slotted spoon to transfer the cooked sausage to a paper towel-lined plate. Leave the sausage's fat in the pan and pour in the egg mixture. Scramble the eggs until almost cooked through but still moist, 3 to 4 minutes. Remove the pan from the heat.

Step 4

Roll and dock the pizza dough. Roll or stretch the pizza dough out into a 12-inch round. Transfer the dough to the baking sheet. Use the tines of a fork to "dock" (poke holes in) the pizza dough, working from the middle out to within 1 inch of the edge. This will prevent the dough from getting soggy from the steaming cheese sauce.

Step 5

Sauce the pizza. Spread on the cheese sauce in a thin, even layer as you would pizza sauce. You might not use all the cheese sauce.

Step 6

Top the pizza. Sprinkle the sausage onto the cheese sauce, followed by the hash browns, and finally the scrambled eggs. Sprinkle the pizza with the mozzarella cheese, remaining 1/4 cup of cheddar cheese, and scallions.

Step 7

Bake for 20 to 25 minutes. Bake until the cheese is melted and and the crust is golden-brown, 20 to 25 minutes.

Step 8

Cool and slice. Let the pizza cool for 10 minutes; this gives the cheese sauce time to set and prevent it from oozing out. Transfer to a cutting board, slice, and serve.