Preheat oven to 425°F. Spray the cut sides of the squash with oil and sprinkle with salt. Place face down on a large baking sheet. Bake until tender and browned on the edges, about 20 to 25 minutes. Meanwhile, in a large saute pan cook sausage on medium heat, breaking up the meat into small pieces as it cooks until the sausage is cooked through and is browned. Add the onion and celery; cook until celery is soft, about 8 to 10 minutes. Add the mushrooms and thyme to the pan, more salt and pepper if needed and cook, stirring 5 minutes, then cook covered for 2 minutes, or until the mushrooms are soft and cooked through. Divide (1/2 cup) between the squash, top with parmesan cheese and bake 10 minutes.