Make the sauce: Place porcini in a small bowl, and cover with 1/2 cup boiling water. Let soak 15 minutes. Lift mushrooms out of water, chop medium fine, and set aside. Strain the liquid through cheesecloth; set liquid aside.
In a large skillet, heat oil and butter over medium-low heat. Add carrot, celery, onion, and garlic; cook, stirring occasionally, until onions are translucent, 3 to 4 minutes.
Add all meats, 1/2 teaspoon salt, and 1/8 teaspoon pepper; raise heat to high, and cook, stirring occasionally, until meat has browned, about 4 minutes. Add wine, and cook until liquid has evaporated, about 2 minutes. Stir in tomatoes, stock, chopped porcini, reserved porcini liquid, and nutmeg. Lower heat to medium low, and slowly simmer, stirring occasionally, about 35 minutes; much of the liquid will evaporate, but the sauce should be very moist, with several tablespoons of liquid remaining. The sauce can be prepared up to 2 days in advance and stored, covered, in the refrigerator.
Make the polenta: Lightly brush a 3-by-5-inch mini loaf pan with oil; set aside. In a medium saucepan, bring 2 2/3 cups water to a boil, and add salt. While constantly whisking, add polenta in a slow, steady stream. When all the polenta has been added, immediately reduce heat to low, and let simmer, stirring constantly with a wooden spoon, 3 to 5 minutes.
Remove polenta from heat, pour into prepared pan, and smooth the surface (an offset spatula works well). Let stand until cool, cover with plastic wrap, and refrigerate until cold and firm, about 4 hours. Can be prepared up to 2 days in advance and stored, covered, in the refrigerator.
Make the bechamel sauce: In a saucepan, melt 2 tablespoons butter over medium-low heat. Add flour, and cook, stirring frequently, 5 to 6 minutes.
Meanwhile, in a small saucepan, bring milk and salt to a boil. Slowly whisk milk into the flour mixture until completely incorporated and smooth. Add nutmeg, and reduce heat to low; simmer, stirring occasionally, until thickened, about 2 to 3 minutes. Remove from heat.
Rub butter over the surface of the bechamel; this will prevent a skin from forming. The bechamel can be prepared up to this point 2 days in advance and stored, covered, in the refrigerator. When ready to use, warm sauce in a small saucepan over low heat, stirring, 1 or 2 minutes. If the sauce is too thick, whisk in about 2 tablespoons of milk to thin it out.
Assemble the lasagna: Heat a medium skillet over medium-high heat; add spinach and a pinch of salt and pepper. Cook, stirring, until wilted and tender, 2 to 3 minutes. Remove from heat, and drain in a colander. Squeeze out all water, finely chop, and set aside.
Unmold polenta onto a clean surface. Using a serrated knife, trim the block of polenta to a length of 4 inches. Turn polenta on its side; slice 6 pieces as thin as possible (they should be less than 1/8 inch thick).
Heat oven to 375 degrees. Spread 5 tablespoons of the meat sauce into each of two 12-ounce ovenproof casseroles. Cover with 1 slice of polenta, and spread 3 tablespoons of meat sauce evenly over. Spread 2 tablespoons of bechamel over the meat sauce, and evenly distribute 2 tablespoons of the chopped spinach over the bechamel. Sprinkle with 2 tablespoons of Parmesan, and cover with 1 slice of polenta. Repeat the layering process (3 tablespoons meat sauce, 2 tablespoons bechamel, 2 tablespoons spinach, 2 tablespoons Parmesan, and 1 piece polenta). Spread 1 heaping tablespoon of bechamel sauce over the final layer of polenta, sprinkle with 1 tablespoon of Parmesan, and dot with 1/4 teaspoon butter. These can be prepared up to this point a day in advance; cover and refrigerate. (If lasagnas are prepared ahead and refrigerated, let them stand at room temperature for 1 hour before baking.)
Bake until tops of lasagnas are golden brown and juices are bubbling, about 20 to 25 minutes. If tops are not yet brown, increase heat to broil and place lasagnas under broiler until tops are speckled brown. Let stand for about 10 minutes before serving.
