Stuffed Delicata Squash with Chicken Sausage-Mushroom Stuffing

small delicata squash (about 16 ounces each), halved and seeded

olive oil spray

kosher salt

sweet Italian chicken sausage

chopped onion

chopped celery

chopped fresh shiitake mushrooms

sprig fresh thyme

shredded parmesan cheese

chopped parsley, for garnish