In a medium saucepan, bring 3 cups water and salt to a boil. Cook broccoli in boiling water, uncovered, for 10 to 12 minutes. Drain, place broccoli in a small bowl, and mash.
Melt butter in a large saucepan over medium heat. Mix in flour, and then gradually whisk in milk and water; stir continuously until liquid begins to boil. Stir in grated onion and mashed broccoli. Season with remaining salt.
