If you can't find Tahitian Pink Banana Squash in your area, butternut or Hubbard squash will work deliciously, too.
Bring water to a boil in a large pot. Add salt and 2 tablespoons of the olive oil. Add the pasta to the boiling water and cook, stirring gently, until al dente, approximately 10 to 12 minutes. Strain, cool with running cold water and toss in the remaining olive oil. Set aside.
Position a rack in the lower third of the oven and preheat to 350 degrees F.
Place the melted butter, sugar, thyme, salt and pepper in a large bowl. Add the squash and toss to coat. Transfer to a parchment-lined baking pan. Roast until tender, about 15 minutes. Let cool and set aside.
In a small saucepan over medium-high heat, melt the 1/2 cup butter, stirring constantly, until the milk solids separate and begin to brown. You will smell a slightly nutty smell when it's properly browned. Remove from the heat immediately and plunge the bottom of the saucepan into an ice bath to stop the cooking process; set aside to cool. Pour off the brown butter and reserve, leaving the sediment in the pan. Reserve the sediment.
In a very large skillet, heat half of the browned butter. Cook the shallots and sage over medium heat until translucent. Add the garlic and cook for 1 minute.
Add the mushrooms and cook until they release their moisture. (If you're using portobellos, be sure to remove the gills or they'll discolor the dish.) Add the sherry and reduce by half.
Add the squash, pasta and remaining browned butter, including the sediment. Cook until the pasta is heated through, then season with salt and pepper to taste. Garnish with fresh sage sprigs and chopped parsley.