Preheat oven to 325°F (163°C). Line a sheet pan with parchment. Split muffins, spread butter on cut sides, and arrange cut side up on the sheet pan.

Heat skillet over medium-high heat, add sausage, and cook until browned, about 5-7 minutes. Remove sausage, leaving drippings in pan.

Lower heat to medium, add butter to drippings. Whisk in flour to create a roux; cook for 2 minutes. Gradually whisk in milk until thickened, about 3-4 minutes. Season with salt and pepper.

Bake muffins for about 10 minutes until golden brown and toasty.

Whisk together eggs, a splash of cream, and salt and pepper. Melt butter in another skillet over medium-low heat. Scramble eggs gently until softly cooked, about 3-4 minutes.

Assemble by ladling gravy on each muffin half, adding sausage, scrambled eggs, and topping with cheddar. Optionally drizzle more gravy.