In a small bowl, add the eggs, milk, salt, and black pepper; whisk the egg mixture together using a whisk. Add one tablespoon of room-temperature butter to the frying pan or large skillet. Over medium-high heat, melt the butter, and then add the eggs, and cook, until the eggs are cooked through, stirring the eggs frequently during the cooking process; remove the eggs from the frying pan and set aside.
Add the slices of ham to the frying pan and cook for about a minute or until they are warm. Remove and set aside. You can also heat them in the microwave or air fryer heating until warm.
With the remaining butter, butter one side of each slice of bread. With the butter side down, add one slice of American cheese, one slice of Swiss cheese, the ham slices, the cooked eggs, American cheese, and then Swiss cheese. Then place the top slice of bread onto the sandwich, butter side up.
Preheat the air fryer to 400 degrees F for 3-5 minutes. Then add your sandwich to the air fryer basket, cook for 2 minutes per side, and remove when the bread is golden brown, and the cheese is melted.
Serve with hashbrowns or shredded hashbrowns.