Prepare the oven and dough: Preheat the oven to 425°F (220°C). Lightly brush a 12-inch (30 cm) pizza pan or large rimmed baking sheet with 1 teaspoon (5 ml) olive oil.

Shape the crust: Stretch or roll the pizza dough into an approximately 12-inch (30 cm) round. Transfer it to the prepared pan and gently press the dough outward, leaving the edge slightly thicker.

Partially bake the crust: Bake the plain crust for 7 to 8 minutes. It should look set and slightly puffed but remain pale in the center. Remove it from the oven and leave the oven at 425°F (220°C).

Brown the sausage: While the crust bakes, place the breakfast sausage in a skillet over medium heat. Cook for 6 to 8 minutes, breaking it into small crumbles, until browned and cooked through to 160°F (71°C).

Reserve the sausage: Transfer the cooked sausage to a plate with a slotted spoon. Leave about 1 tablespoon of sausage drippings in the skillet. If very little fat remains, add the 1 tablespoon (14 g) butter.

Make the gravy base: Sprinkle the flour over the fat in the skillet. Cook over medium heat for 1 minute, stirring constantly, until no dry flour remains.

Add the milk: Gradually whisk in the 1 cup (240 ml) whole milk. Add the salt and black pepper. Cook for 3 to 5 minutes, whisking frequently, until the gravy is smooth and thick enough to spread without running.

Finish the sausage gravy: Stir half of the cooked sausage into the gravy. Remove the skillet from the heat. The gravy should coat the spoon but still spread easily across the crust.

Whisk the eggs: In a medium bowl, whisk the eggs with the 2 tablespoons (30 ml) milk until no distinct streaks of egg white remain.

Soft-scramble the eggs: Melt the remaining 1 tablespoon (14 g) butter in a clean skillet over medium-low heat. Add the eggs and cook for 2 to 3 minutes, stirring slowly, until they form soft curds and are mostly set but still slightly glossy. Remove them from the heat before they look completely firm.

Add the gravy: Spread the sausage gravy over the partially baked crust, leaving approximately a 1/2-inch (1.25 cm) border around the edge.

Add the eggs and sausage: Scatter the softly scrambled eggs evenly over the gravy. Sprinkle the remaining cooked sausage over the eggs.

Add the cheese: Sprinkle the shredded cheddar evenly over the pizza, extending it almost to the crust edge.

Bake the breakfast pizza: Return the pizza to the 425°F (220°C) oven and bake for 10 to 12 minutes. The cheese should be completely melted, the crust edges should be golden brown, and the eggs should reach 160°F (71°C).

Rest before slicing: Remove the pizza from the oven and let it rest for 5 minutes. This allows the gravy and cheese to settle and makes the slices easier to lift.

Garnish and serve: Sprinkle with sliced chives, if using. Cut the pizza into 8 slices and serve warm with hot sauce as desired.