Cut broccoli into florets. Chop onion. Prepare cooked spaghetti. Set aside. Steam cook broccoli. Set aside. Cut cheeses into 1 cm pieces.
In a small saucepan, melt butter. Add onion and cook until translucent. Mix in flour, salt, and pepper and cook for about 1 minute. Gradually add milk, stirring constantly. Increase heat to medium high. Keep stirring until mixture thickens and bubbles, then reduce heat to medium, now stirring about every two minutes. Add Provolone and mozzarella cheeses.
Pour sauce mixture over spaghetti. Mix together with broccoli.