Whisk egg and milk together and pour into a large skillet set to medium heat. Cover with a lid.

every couple of minutes gently "swirl" the pan around to move the liquid to the outer rim of the skillet, then run a rubber spatula around the edge. Cover and continue to cook until there is no liquid left. Using a glass that is about the same size as your sausage patty, cut circles of eggs out of the skillet, gently picking them up and placing them in a baking dish, dividing the layers with waxed paper. Refrigerate until ready to use.

Grill the sausage patties until cooked through. Top with egg, then cheese. Heat until cheese is melted and egg is hot.

Serve on croissants.