Heat a 9 inch cast iron skillet over medium heat. Melt butter. Add potatoes and cook, stirring frequently, until translucent and tender (mine took about 8 minutes, but it will depend on how small you cut them). Add diced ham to the potatoes and cook another 2 minutes.
In a bowl, whisk eggs, milk, 1c of cheese, green onions, and salt and pepper (ham adds a lot of salt, so you won't need much, maybe 1/4t). Pour into skillet, give a quick stir, cover, and reduce heat to low - 6 minutes.
Meanwhile, preheat broiler. Remove lid from pan and top with remaining cheese. Transfer skillet to broiler and broil until eggs are set (no longer jiggling and firm to the touch), cheese is melted, and top begins to turn a nice golden brown. Garnish with more green onion. Slice into six wedges and serve.