1. In a bowl, combine mushrooms and 1/2 cup boiling water and let stand until limp, about 5 minutes.
2. Meanwhile, cut the polenta crosswise into 1/2-inch-thick slices. Divide slices among 4 shallow 2-cup gratin dishes or ramekins. Bake in a 450° oven until hot, about 5 minutes.
3. As polenta bakes, rub mushrooms gently to release grit, then lift from water, squeeze dry, and finely chop. Reserve liquid.
4. Put mushrooms in a 10- to 12-inch frying pan. Pour in most of the soaking liquid, discarding the gritty remainder. Add tomatoes, wine, and basil. Bring to a boil over high heat and boil about 5 minutes, stirring often.
5. Remove polenta from oven and spoon sauce evenly over it. Sprinkle with jack and parmesan cheeses.
6. Broil 6 inches from heat until cheeses begin to brown, about 6 minutes.