Preheat oven to 400 degrees and spray a muffin pan well with non-stick spray or line with paper liners.
In a bowl, whisk eggs well and season with salt and pepper.
In a skillet, melt butter and add eggs. Cook and stir over medium heat until eggs are thickened and no liquid egg remains. Be sure to break up the eggs so they will be distributed through all the muffins.
In a large bowl, prepare the corn muffin mix according to the package directions, using the additional egg and the 1/3 cup milk.
Gently stir the sausage, scrambled eggs, and cheese into the corn muffin batter.
Fill muffin cups 2/3 full.
Bake 15 minutes, or until a toothpick inserted in center comes out clean.