Preheat oven to 400ºF (200ºC).
Place squash halves cut sides up on a baking sheet and brush the entire open surface of each half with olive oil. Season with salt and pepper and roast for 30 minutes.
While the squash is roasting, brown the sausage in a skillet over medium heat breaking it up as you go, about 5-7 minutes. Add the onion to the pan and continue to cook for another 5 minutes until the onion has softened.
Turn off the heat and stir in balsamic vinegar, garlic, Parmigiano Reggiano and thyme leaves.
Remove squash from oven and fill each half with the sausage mixture, then roast for another 20 minutes or so until the squash is tender when pierced with a fork.
Top with more Parmigiano Reggiano and thyme leaves as desired. Serve hot.