Bring chicken stock to a boil and cook broccoli until crisp tender.
In a saucepan add 2 tbsp. butter and saute onion, garlic, and mushrooms 5 min. Set aside.
In separate saucepan add the other 4 tbsp. butter stir in flour and make a roux. Add milk slowly whisking to combine. Whisk in egg. Stir in 2 cups of the cheese saving the other 1/2 cup for topping. When cheese mixture is smooth and cheese is melted stir in onion mixture.
Bake at 350 for 30 minutes. Top with the reserved 1/2 cup cheese and bake 10 minutes longer until cheese is melted