Preheat your oven according to the pizza crust or biscuit dough package directions. If using biscuit dough, press it into a 12-inch pizza shape on a baking sheet.
Cook the breakfast sausage in a skillet over medium heat, breaking it up until fully browned. Remove the sausage with a slotted spoon, leaving about 2 tablespoons of drippings in the pan.
Add the butter to the drippings in the skillet. Once melted, whisk in the flour and cook for 1 minute, stirring constantly.
Gradually whisk in the 2 cups of whole milk until smooth. Bring the mixture to a simmer, stirring often until the gravy thickens enough to coat the back of a spoon, about 5 to 7 minutes.
Stir the cooked sausage, salt, black pepper, and cayenne pepper (if using) into the gravy. Remove from heat.
In a separate bowl, whisk the 4 eggs with the 1/2 cup of milk until combined. Pour the egg mixture into the skillet used for the gravy (no need to clean it). Scramble the eggs over medium-low heat until just set.
Spread the sausage gravy evenly over the prepared pizza crust.
Spoon the scrambled eggs over the gravy layer.
Sprinkle the shredded cheddar cheese over the eggs and gravy.
Bake according to the crust instructions, usually 12 to 18 minutes, until the crust is golden brown and the cheese is melted and bubbly.
Slice and serve immediately.
Notes
You can substitute the pre-made crust with homemade pizza dough or use refrigerated crescent roll dough for a flakier base.
For a richer gravy, use heavy cream instead of some of the whole milk.
Prepare the sausage gravy ahead of time and store it in the refrigerator for up to 3 days. Reheat gently before assembling the pizza.