Melt the butter in a 10-inch oven-safe skillet over medium heat. Sauté the onion and broccoli until the broccoli is bright green and just tender, about 5 minutes. Add the garlic and cook 30 seconds, then remove from heat.

Whisk the eggs, milk, salt, and pepper until smooth. Stir in half of the cheddar.

Return the skillet to medium heat, pour in the egg mixture, and scatter the remaining cheddar on top. Cook without stirring until the edges are set and the center is still jiggly, about 5 minutes.

While the frittata cooks, preheat the broiler with a rack 6 inches from the heat. Transfer the skillet to the broiler and broil until puffed and golden, 3 to 4 minutes. Rest for 2 minutes, then slice and serve.