Prepare an outdoor grill for cooking over medium heat. Line a baking sheet with parchment paper.
Heat the oil in a large nonstick skillet over medium-high heat. Add the sausage and cook, stirring and breaking up with a wooden spoon, until browned and cooked through, 8 to 10 minutes. Remove from the heat, and use a slotted spoon to transfer the sausage to a plate.
Reserving 2 tablespoons of drippings in the skillet (drain or add oil as needed), return it to medium-high heat. Whisk in the flour and cook, whisking frequently, until the mixture becomes loose and turns a light tan color, 1 to 2 minutes. Slowly whisk in the milk, 1/4 teaspoon salt and 1/2 teaspoon pepper. Simmer, whisking frequently, until the sauce has thickened to the consistency of thin gravy, 6 to 8 minutes. Return the cooked sausage to the skillet and stir. Remove from the heat, cover and set aside.
Divide the pizza dough into 4 balls. Dust a clean flat work surface with flour, and roll each ball into an 8-inch circle. Transfer the dough circles to the prepared baking sheet, and brush both sides with oil.
Lightly oil the grill grates. Grill the crusts until they start to puff and distinct grill marks appear on the underside, about 1 minute. Flip. Evenly divide the Cheddar among the crusts, leaving a small well empty in the center of each. Immediately crack 1 egg into each well. Grill, covered, until the whites have completely set and the Cheddar has melted, 8 to 10 minutes.
Meanwhile, reheat the sausage gravy if needed. Remove the pizzas from the grill and drizzle each with a couple spoonfuls of warm gravy. Garnish with the scallions. Serve immediately with hot sauce and any remaining gravy served on the side.