Preheat the oven to 350 degrees. Grease a 9-inch baking dish with non-stick cooking spray. In the baking dish, toss together gruyere cheese, ham, and broccoli. In a mixing bowl, whisk together eggs, milk, and seasonings. Pour mixture over ham and gently move ham and cheese until the egg mixture is evenly distributed. Bake for 45 minutes or until the eggs are completely set. Remove from the oven and allow the quiche to rest for 5 minutes before slicing and serving.