01 - Preheat your oven to 425°F (220°C). Line a baking sheet or pizza stone with parchment paper for easier handling. Lightly flour a clean surface and roll out the pizza dough to approximately 12 inches in diameter. Place the rolled dough onto the prepared baking sheet.
02 - In a large skillet over medium heat, cook the breakfast sausage until it is fully browned and crumbled. This should take about 5 to 7 minutes. Using a slotted spoon, transfer the cooked sausage to a separate dish, leaving approximately 2 tablespoons of drippings in the skillet. Add the 2 tablespoons of unsalted butter to the drippings and allow it to melt. Once melted, whisk in the all-purpose flour and cook for 1 minute until it forms a bubbly paste.
03 - Gradually whisk in the whole milk, ensuring a smooth consistency without lumps. Bring the mixture to a simmer and continue to stir constantly until the gravy thickens, which usually takes about 3 to 4 minutes. Stir in the cooked sausage, salt, and black pepper. Remove the skillet from the heat.
04 - In a separate bowl, whisk together the 4 large eggs and 2 tablespoons of milk. Melt the remaining 1 tablespoon of unsalted butter in a nonstick skillet over medium heat. Pour in the egg mixture and scramble gently until the eggs are just set but still soft in appearance. Remove from heat.
05 - Spread the prepared sausage gravy evenly over the surface of the pizza crust, leaving a small border around the edges. Scatter the scrambled eggs uniformly over the gravy. Top generously with both the shredded cheddar and mozzarella cheeses.
06 - Transfer the assembled pizza to the preheated oven and bake for 12 to 15 minutes, or until the crust achieves a golden-brown hue and the cheese is completely melted and bubbly. Remove the pizza from the oven and allow it to cool for 2 to 3 minutes. Garnish with chopped fresh chives or green onions, if using. Slice and serve immediately while warm.