Divide sausage into 4 equal portions and shape into patties slightly larger than your English muffins (they'll shrink while cooking).
Cook sausage patties in a large skillet over medium heat for 4-5 minutes per side until browned and cooked through (internal temperature 160°F). Transfer to a plate and cover with foil to keep warm.
Whisk together eggs, milk, salt, and pepper in a bowl. Melt butter in the same skillet over medium-low heat.
Pour in egg mixture and cook, stirring gently, until soft scrambled eggs form, about 3-4 minutes.
Place sausage patties on the bottom halves of the toasted English muffins. Top each with a slice of cheese, then divide scrambled eggs evenly among the sandwiches.
Cap with the top halves of the English muffins and serve immediately while hot and melty.