Special equipment: a spiralizer
Preheat the oven to 400 degrees F.
Using a spiralizer, make "buttles" (butternut squash noodles) from the squash.
Heat 2 tablespoons olive oil in a medium saute pan over medium heat until glistening. Add the squash noodles and saute until softened, 7 to 10 minutes. Season with salt and pepper.
Pulse 1/2 cup of the chanterelles and 1/2 cup of the king trumpet mushrooms in a food processor until ground.
Heat 4 tablespoons olive oil in a medium saute pan over medium heat. Fry the sage until crispy, then transfer to a paper towel-lined plate. Add the garlic and shallots to the pan and sweat until just translucent, about 3 minutes. Add the ground mushrooms and cook until starting to brown, about 7 minutes. Season with salt and pepper. Add the cream and wine and bring to a boil. Reduce the heat to low and simmer until the liquid is reduced by half, about 10 minutes.
On a baking sheet, spread out the remaining 1/2 cup chanterelles, 1/2 cup king trumpets and the Brussels sprouts in a single layer. Drizzle with olive oil to lightly coat; season with salt and pepper. Roast until starting to brown and the Brussels sprouts are tender, about 15 minutes.
Toss the noodles in the cream sauce, then divide among four plates. Top with the roasted Brussel sprouts and mushrooms. Sprinkle with the Parmesan and garnish with the crispy sage.