Bake the squash: Preheat the oven to 400°F. Cut the squash in half and scoop out the stringy core and seeds. Arrange the halves cut-sides up on a rimmed baking sheet. Brush each half with 1 teaspoon olive oil and sprinkle with salt and pepper. Bake for 40 minutes, until the edges begin to wrinkle and the flesh is fork-tender. Set aside. Reduce the oven temperature to 375°F.
While the squash bakes, heat the olive oil in a large skillet over medium-high heat and add the sausage. Brown the meat, breaking it apart into small pieces for 3 minutes.
Add the mushrooms, onion, apple, salt, pepper, nutmeg, allspice, and minced garlic. Cook, stirring frequently, until the sausage is cooked through, the vegetables and apple are softened, and the liquid has cooked off, about 8 minutes.
When the squash is cool enough to handle, scoop out its flesh, leaving a wall all the way around that is about ¼-inch thick. Add the scooped squash to the pan with the sausage. Add the herbs and ¼ cup Parmesan. Stir to combine. Taste and adjust salt and pepper as desired.
Mound the filling inside of the hollowed squash halves. Sprinkle the remaining Parmesan over the top and bake until the cheese is melted and the filling is hot, about 15 minutes. Sprinkle with additional fresh herbs as desired and serve.
