Brown and crumble the sausage in a non-stick skillet over medium heat, 6–8 minutes. Transfer to a paper towel-lined bowl to drain.

Wipe the skillet. Melt 2 tbsp butter over medium-high heat. Pour in eggs, let sit briefly, then gently stir until just set, about 3–4 minutes. Remove from heat.

Discard the paper towel. Add eggs to sausage and gently stir to combine.

Butter one side of each bread slice. Place one slice butter-side down in the skillet, top with cheese, ½ cup egg-sausage mix, another cheese slice, and finish with a second bread slice, butter-side up.

Cook over medium heat for 3–4 minutes per side until golden brown and the cheese is melted.

Rest sandwiches for 1 minute, then slice and serve warm.