Preheat the oven to 350 degrees F. Grease a 9 by 9 by 2-inch square baking dish with the butter. In a mixing bowl, whisk the eggs and milk together. Season the mixture with salt and pepper. Add 1 cup of the Cheddar cheese, 1 cup of the White Cheddar cheese, the ham and the tater tots. Mix the mixture thoroughly. Pour the mixture into the prepared pan and sprinkle the top with the remaining cheese. Place in the oven and bake for 40 to 45 minutes or until the top is brown.