Preheat the oven to 400 degrees F.
Place the squash on a small baking sheet and toss with 1 tablespoon of the oil. Roast until easily pierced with a fork, 20 to 30 minutes. Set aside.
Place the cornbread and Italian bread on a baking sheet in single layer. Bake until light brown, 10 minutes. Set aside and let cool.
Remove the tops of the pumpkins, and reserve. Clean out the insides of the pumpkins by removing the seeds and scraping the cavities with a spoon. Set the pumpkins aside.
In small pot, on medium heat add the figs and 1 cup of the stock. Bring to a simmer and cook until the figs absorb all of the stock, 6 to 8 minutes.
Melt the butter in a large stockpot over medium heat. Add garlic, celery, leeks, onions, thyme, 1 teaspoon salt and 1/2 teaspoon pepper. Cook until translucent and soft, 8 to 10 minutes. Let cool.
To the stockpot, add the toasted cornbread and Italian bread, the roasted squash, the fig mixture and the parsley; gently toss. In small bowl, whisk 1 egg with the remaining 1 1/2 cups stock; pour over the stuffing and toss until all the liquid is absorbed. (The stuffing can be made a day ahead at this point; just cover in plastic wrap and keep in refrigerator.)
Fill each pumpkin to the top with a heaping 1/2 cup of the stuffing. Place the filled pumpkins with the tops alongside in a large roasting pan. Cover with foil and bake for 20 to 25 minutes. Remove the foil, sprinkle the pumpkins with the Parmesan and bake for another 5 minutes, or until stuffing is lightly browned.
Remove the pumpkins from the oven and let sit for 5 minutes before serving. Garnish with thyme sprigs and serve with the pumpkin tops resting against the sides of the pumpkins. Serve.
The stuffing can also be baking on its own. Grease a 9-by-13-inch baking pan with 1 tablespoon butter, place stuffing mixture in the pan, cover with foil, and bake for 40 minutes. Remove the foil for last 10 minutes to lightly brown.