Set an oven rack about 6 inches from the heat source; preheat the broiler.

Beat eggs and milk in a bowl; set aside.

Melt butter in a large, oven-proof skillet over medium heat; cook and stir potatoes, ham, onion, and garlic in melted butter until potatoes are tender, about 15 minutes. Add water if vegetables become dry. Season with salt and pepper.

Pour egg mixture over vegetable mixture. Cook, without stirring, until eggs are slightly browned on the bottom.

Broil in the preheated oven until eggs are nearly set, about 5 minutes. Sprinkle Cheddar cheese over eggs and continue broiling until frittata is completely set and cheese is melted, about 5 minutes more.