To begin to prepare the sauce, in a small bowl combine the dried porcini and hot water and let stand for 45 minutes to 1 hour. Remove the porcini and squeeze out excess water. Using scissors, cut them into pieces about the size of your thumbnail. Set aside. Strain the mushroom liquor through a paper towel or a fine sieve; reserve. Lightly oil an 8-by-11-inch baking pan and set it aside. Lightly oil two large pastry boards or baking sheets, or a very large scratchproof counter surface. Make the polenta and when it is cooked, turn it directly out onto the boards, counter, or baking sheets. Use a large knife or rubber spatula, first dipped in cold water, to spread out the polenta to a thickness of about 1/2 inch. Allow it to cool and harden, about 20 minutes. Cut it into approximate 3-inch squares.

Meanwhile, preheat an oven to 400 degrees F. Continue to make the sauce. Using a soft brush or clean kitchen towel, remove any dirt from the fresh mushrooms. Don't wash them because water will alter their texture. Separate the stems from the caps, discarding the stems if they are tough. Slice the mushroom caps and tender stems thinly. In a skillet over medium-low heat, melt the butter with the oil. Add the onion and saute until softened, about 5 minutes. Add the porcini and saute for another 5 or 6 minutes to marry the flavors. Add the cultivated mushrooms and continue to saute until tender, about 5 minutes. Add the mushroom soaking liquid, salt, and pepper. Allow to simmer gently for 5 minutes, stirring occasionally. Then stir in the cream, and allow the mixture barely to reach a simmer. Immediately remove the skillet from the heat.

Place half of the polenta squares in a layer on the bottom of the prepared dish. Spread half of the sauce in a layer over them, then sprinkle with half of the cheeses. Place another layer of polenta squares over the cheeses. Spoon the remaining sauce on top and sprinkle with the remaining cheeses. Place on the middle rack of the oven and bake until the cheese is melted and golden and the pasticcio is bubbling, about 20 minutes. Remove from the oven and let settle for 10 minutes before cutting into squares. Serve hot.

Ahead-of-time note: This pasticcio can be assembled up to 3 days in advance, covered, and refrigerated, and then baked just before serving. Leftover pasticcio reheats well: Cover with aluminum foil and place in a preheated 375 degree F oven for 20 to 30 minutes, depending on whether it is chilled or at room temperature.