Preheat oven to 350°F. Coat a 2-quart rectangular baking dish with cooking spray.
Combine 2 cups broccoli florets and 2 tablespoons water in a medium microwave-safe bowl. Cover and microwave on High until crisp-tender, 1½ to 2 minutes. Drain and rinse with cold water. Drain again and pat dry.
Combine 4 cups thawed hash brown potatoes, the ham pieces, 3 tablespoons chives and the cooked broccoli in the prepared baking dish. Toss to combine; spread in an even layer.
In a medium bowl, combine 6 eggs, ¾ cup milk, 1 teaspoon garlic powder, ½ teaspoon salt and ½ teaspoon pepper; whisk until smooth. Pour over the potato mixture. Sprinkle evenly with 1 cup cheese. Cover the dish with foil.
Bake until the cheese is mostly melted, about 30 minutes. Remove the foil and continue baking until the cheese is browned and the eggs are set, about 30 more minutes. Let stand for at least 5 minutes. Garnish with additional chives, if desired.
