Sprinkle quail with 2 teaspoons salt and 1/2 teaspoon pepper. Dredge in 2 cups flour. Heat oil to 400°, and brown quail on both sides in a large skillet. Pour water over quail. Reduce heat; cover, and cook 10 minutes. Uncover, and simmer an additional 35 minutes. Remove quail to serving platter. Drain off drippings, reserving 6 tablespoons in skillet.
Add remaining flour to reserved drippings; cook over low heat, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add milk; cook over medium heat, stirring constantly, until thickened and bubbly. Stir in remaining salt and pepper. Serve gravy with quail.