In a bowl stir together turkey, Worcestershire sauce, and mustard and form into two 3/4-inch-thick burgers. Season burgers with salt and pepper.
In a large heavy skillet heat 2 teaspoons oil over moderately high heat until hot but not smoking and sauté burgers until cooked through, about 6 minutes on each side. Transfer burgers to a plate.
To skillet add onion and remaining teaspoon oil and cook over moderate heat, stirring, 3 minutes. Add mushrooms and sage and cook, stirring, until liquid mushrooms give off is evaporated. In a small bowl whisk together broth and flour until flour is dissolved and stir into mushrooms. Simmer gravy, stirring, 5 minutes and spoon over burgers.
