Polenta topped with mushrooms is a savory staple, so we thought we'd turn it upside down.
In a small bowl, combine porcini mushrooms with boiling water. Let stand for 15 minutes. Pass soaking liquid through damp cheesecloth into a small bowl, and set aside. Finely chop porcini mushrooms, and set aside.
Melt butter in a large saute pan over medium heat. Add garlic and shallot, and saute until tender, 3 to 4 minutes. Raise heat to medium high, and add button mushrooms; cook for 2 minutes. Add remaining mushrooms, and cook until tender, about 5 minutes.
Stir in flour, and cook 1 minute. Stir in red wine, using a wooden spoon to scrape up any cooked-on bits on bottom of pan. Allow the liquid to reduce for 1 minute. Stir in tomato paste and the reserved porcini liquid, and simmer for 2 minutes. Stir in heavy cream, parsley, and thyme; season with salt and pepper to taste.
Heat oven to 375 degrees. Combine 2 cups milk with 2 cups water in a large saucepan, and bring to a simmer. Add polenta, and cook, stirring, until thickened, 1 to 2 minutes. Remove from the heat, and stir in Parmesan, remaining 1 1/2 cups milk, and 1 teaspoon salt. Place egg yolks in a medium mixing bowl, and stir in 1 cup of the polenta mixture. Stir the egg-and-polenta mixture back into the polenta, and adjust seasoning with salt, if necessary.
In a large mixing bowl, beat egg whites until stiff peaks form. Stir half of the egg whites into the polenta, then gently fold in the remaining whites until completely incorporated. Set aside.
Place the mushroom mixture in a shallow 2-quart casserole; heap polenta on top. Sprinkle with Parmesan, and place on a baking sheet. Bake until polenta is golden and mushroom filling bubbles, about 25 minutes.