To a small non reactive bowl or dish combine marinade ingredients. Marinate tofu overnight.
Place tofu slices on hot grill and cook 4 to 5 minutes on each side. Place grilled slices in a warm place while making the sauce.
In a separate saucepan whisk together olive oil, milk, flour and vegetable broth. Add shiitake mushrooms and cook over medium heat, stirring occasionally, for 7 to 10 minutes or until the mushrooms are tender. Pour the mixture over grilled tofu. Serve immediately.