Preheat the oven to 350°F (175°C).
In a skillet, melt the butter over medium heat.
Sauté the diced ham for about 3-4 minutes until heated through.
In a bowl, whisk together the eggs, milk, black pepper, and half of the cheese.
Pour the egg mixture over the ham in the skillet, ensuring even distribution.
Cook on the stovetop for 2-3 minutes until the edges begin to set.
Sprinkle the remaining cheese on top before transferring to the oven.
Bake in the preheated oven for 15-20 minutes, or until the eggs are fully set.
Once baked, remove from the oven and let it cool for a few minutes.
Garnish with chopped green onions if desired and serve warm.
