Heat a skillet over medium heat and melt the butter.
Add the ham and sauté until browned.
Meanwhile, in a large bowl, beat together the eggs, milk, ½ teaspoon salt, and ¼ teaspoon pepper. Pour into the pan and cook, and stir occasionally, to your desired taste approx. 4 to 5 minutes
Add grated cheddar and chives just before the eggs begin to set.