Heat a charcoal grill with coals. Spread them out in one dense layer.
Whisk the eggs in a bowl with the milk, salt, and pepper. Heat 2 tablespoons of butter in a large saute or omelet pan over the hot grill. Add the eggs and cook them until the desired doneness, stirring constantly. Off the grill, add the goat cheese, chives, and the remaining 2 tablespoons of butter. Stir and allow the eggs to sit for 30 seconds, until the cheese begins to melt. Check for seasonings. Serve hot.