Preheat broiler. In a 10-inch (25 cm) non-stick skillet, melt butter over medium heat. Add potatoes, green onions and peppers; cook for 10 to 12 minutes or until tender, stirring often.
Meanwhile, in a large bowl or glass measuring cup, whisk eggs, milk, mustard and pepper; stir in ham and pour over potato mixture. Cook, until eggs are almost set but still moist, about 10 minutes, occasionally lifting edges with a spatula and tilting pan to allow uncooked egg to flow underneath. Sprinkle with cheese.
Broil until golden brown and cheese is melted, about 2 minutes. Run a spatula around the outside and invert onto a serving plate; cut into wedges.
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