Heat sausage in a large skillet over medium-high heat. Cook, crumbling with a spoon and stirring often, until browned and cooked through, 5 to 7 minutes.

Add butter to skillet and melt. Sprinkle flour evenly over skillet and stir for 1 minute. Whisk in 2 1/2 cups milk and scrape up any bits off the bottom of the skillet. Bring mixture to a simmer.

Reduce heat to low and cook, stirring often, until thickened, about 10 minutes. If mixture becomes too thick, thin slightly by adding more milk or water. Season with hot sauce, pepper, and salt; cook for 1 more minute.

Serve immediately over hot biscuits.