Heat the butter and olive oil in heavy, large pot over medium-high heat. Add the onions and saute until tender, about 7 minutes. Add the wild mushrooms and saute until golden brown and tender, about 8 minutes. Add the thyme and garlic and saute 2 minutes. Add the mushroom broth and cream and bring to a boil. Reduce the heat to medium and simmer gently until the soup thickens slightly, about 12 minutes. Transfer the soup to a blender and in small batches, puree until smooth. Return to the pot. Bring the soup to simmer. Season with salt and pepper. To plate the soup, place a tablespoon of the Kuri Squash Puree in the center of a large soup plate. Then top the squash with the Chestnut Mushroom Raviolis. Ladle the cream of mushroom soup around the raviolis and top with the Maitake Mushroom Fritters.
Preheat the oven to 350 degrees F. Toss the squash in a medium bowl with the olive oil and thyme sprigs. Season with salt and pepper. Place the squash pieces on a baking sheet and bake until the squash is very soft, about 40 minutes. Then, using a spoon, scrape the squash off the skin and place the flesh of the squash into a blender with the butter and blend until smooth.
For the pasta dough: Blend the flour, eggs, yolks, pinch salt and 2 tablespoons water in a food processor until the mixture just begins to form a ball, adding more water drop by drop if the dough is too dry. Process the dough for 15 seconds more to knead it. Wrap in plastic wrap and let stand for 1 hour to let the gluten relax.
For the filling: In a large pot, heat the butter and olive oil over medium-high heat. Add the garlic and shallots and cook for 3 minutes. Then add the mushrooms, seasoning them with salt and pepper. Cook until all the liquid has evaporated. Finish with the chopped parsley. Let the mixture cool before filling the ravioli. To form the ravioli, divide the dough in half and roll one part through the thinnest setting of a pasta machine. Avoid flouring the pasta dough during the rolling unless absolutely necessary. Cut the strip of pasta into 2 foot sections. Working on a lightly floured table, take one section at a time and fold it in half lengthwise. Unfold it and use the center crease as a guideline to add the mushroom filling. Put 1 teaspoon filling every 2 inches just under the centerline of one half of the strip of pasta. Spray the pasta with a fine mist of water or use a pastry brush to lightly moisten the dough between and below the filling. Fold the top half of the dough over the filling and the bottom half of the dough, pressing firmly between the filling. Lightly flour the strip of ravioli. Then, using a square cutter, cut off the uneven bottom end along the strip, and then cut in between each dot of filling to form the individual ravioli. Set the ravioli on a lightly floured baking sheet and keep in the refrigerator. Repeat the process until all of the filling and dough are used. Bring a large pot of salted water to a boil. Gently drop the raviolis into the water. Cook for 3 minutes. Remove the raviolis with a slotted spoon onto a dinner plate. Drizzle lightly with olive oil to avoid sticking together.
In a medium mixing bowl, combine the ice and 1 cup cold water and let sit for 5 minutes. Measure out 1 cup of ice water into a bowl. Using a fork, stir in the flour and egg yolk into the water until the batter is smooth. In a large, heavy-bottomed pot, heat the oil over medium-high heat. Then, in small batches, dip the mushrooms into the batter and carefully drop into the hot oil. Remove the fritters with a slotted spoon when they are golden brown onto a plate with paper towels. Season with salt.
When blending hot liquids, remove liquid from the heat and allow to cool for at least 5 minutes. Transfer liquid to a blender or food processor and fill it no more than halfway. If using a blender, release one corner of the lid. This prevents the vacuum effect that creates heat explosions. Place a towel over the top of the machine, pulse a few times then process on high speed until smooth.