Brown hash browns and ham in 1 tablespoon of oil in a medium skillet over medium high heat. Stir in diced green chilies and 1/2 cup of Cheddar cheese. Cook until cheese has melted.
Coat the bottom of a 9x13 inch baking dish with a small amount of enchilada sauce. Dip each tortilla in remaining sauce, and fill with potato-ham mixture. Roll each as tightly as possible and place in the baking dish, seam side down. Top with remaining sauce and cheese, and cover with tin foil.
Bake, covered, in the preheated oven for approximately 20 minutes. Remove foil and bake for another 10 minutes, or until lightly browned on top. Serve immediately.