Trim off large leaves of broccoli; remove tough ends of lower stalks. Wash broccoli. Remove flowerets. Cook, covered, in a small amount of boiling salted water 5 minutes; drain and set aside.
Sauté onion in oil in a 10-inch skillet until tender. Place broccoli over onion.
Combine eggs, milk, garlic salt, and pepper, mixing well. Pour egg mixture evenly over broccoli. Sprinkle cheese evenly over egg mixture. Cook, covered, over low heat 25 minutes or until set. Cut into wedges, and serve immediately.