In a large nonstick skillet, over medium heat, melt the butter. Saute the ham until browned.
Meanwhile, in a large bowl, whisk together the eggs, the milk or water, the salt, and the pepper. Pour the mixture into the pan and cook, stirring occasionally, to desired doneness, 4 to 5 minutes. Just before the eggs are set, stir in the Cheddar and the chives.