In a mixing bowl whisk together the beaten eggs with kosher salt and white pepper.

Warm a large skillet over medium heat and melt 2 tablespoons of butter. Spread the remaining tablespoon of butter on one side of each bread slice and set aside.

Pour the eggs into the skillet and gently tilt the pan so the mixture evenly covers the surface. Position both slices of bread, buttered sides facing upward and edges touching, directly atop the eggs. Allow the eggs to cook undisturbed for 5–7 minutes until set.

Carefully flip the eggs along with the bread slices in one motion.

Place the cheddar cheese slice and ham atop one bread slice, folding any excess cooked egg over the ham. Close the sandwich by folding the other slice of bread over the fillings.

Continue to cook for an additional 2–3 minutes, flipping as needed, until the cheese has melted and the sandwich is golden and warmed through. Serve immediately while hot.