Bring onion and water to a boil in a saucepan. Cover, reduce heat, and simmer until tender, 15 to 20 minutes. Transfer onion and water to a blender; puree until smooth.
Heat oil in a saucepan over medium heat. Whisk in flour to form a thick paste the consistency of cake frosting. Continue cooking, whisking constantly, until flour has turned golden and begins to smell like toast, about 20 minutes.
Stir onion puree into flour paste. Cook and stir until thickened to sauce consistency, adding more water if necessary, about 2 minutes. Stir in wine vinegar and salt.
Onion can also be cooked in water in microwave on high until tender, about 10 minutes.
Do not omit or substitute wine vinegar with any other type of vinegar. Cooked onions taste somewhat sweet, and wine vinegar gives the sauce just the right taste balance. Alcohol, cider or balsamic vinegar are not a good option because they would completely change the distinctive taste of the sauce.