Preheat the oven to 425 degrees F. Grease 2 custard cups or ramekins with the butter
Divide the thawed hash browns between the 2 dishes. Firmly press into the bottom and partway up the sides. Place in the preheated oven and bake for 15 minutes, the hash browns should be starting to brown.
While the potatoes are baking place the olive oil into a small skillet and heat over medium heat. Add the onions and pepper and saute for 2 to 3 minutes.
Add the chopped ham to the skillet and continue to saute for another 2 minutes. The onions and peppers should be softened and the ham lightly browned. Remove from the heat and set aside.
When the ham mixture is done prepare the eggs. Whisk the eggs with the cheese and set aside. Season with salt and pepper to taste.
Once the potatoes are baked remove from the oven. Divide the ham mixture between the 2 dishes. Then divide the egg between the 2 dishes by pouring them over the ham mixture.
Place back in the oven and bake for another 17 to 19 minutes. Remove from the oven and let them sit for a few minutes. Remove from the dishes and place them on plates and serve.