Bring water and salt to a boil in a small saucepan; cook onions in the boiling water until just tender, 5 to 10 minutes. Drain, reserving 1/2 cup water in saucepan, and return onions to pan. Add butter to onions.
Mix milk and quick-mixing flour together in a bowl; add to onion mixture. Cook and stir over medium-low heat until thickened, 10 to 15 minutes. Stir in more quick-mixing flour if mixture is too thin.