Preheat oven to 350 degrees F. Place the eggs in a bowl and beat until yolks and whites are well combined. Mix in the milk, salt, pepper and nutmeg, and then set aside. Distribute half the cheese, in small pieces, into the bottom of the pie shell. Arrange the broccoli on top of the cheese. Top the broccoli with the remaining cheese; sprinkle with the red bell pepper. Bake the quiche 45 to 50 minutes, or until the eggs are set. Rest the quiche for 10 minutes before cutting and serving.
Note: To blanch broccoli, plunge into boiling water for 2 minutes. Cool in cold water, drain well, and pat dry.