Preheat the oven to 400°F. Line a large rimmed baking sheet with aluminum foil and lightly coat with nonstick cooking spray.
Carefully cut and remove the top and bottom ends of the squash. Slice in half down the middle so the squash halves look like a flower, and scoop out the seeds. (Another way to slice the squash is to cut it in half from stem to tip.)
Brush the insides of the hollowed out squash with 2 teaspoons of the olive oil, and season with a few pinches of salt and pepper. Place, hollowed side down, on the prepared baking sheet. Place in the oven and bake until tender, 35 to 40 minutes.
While the squash is in the oven, heat the remaining 1 tablespoon of oil in a large nonstick skillet over low heat. Add the onion, mushrooms, and garlic and cook, stirring frequently, until the vegetables are tender, about 7 minutes. Add the turkey and thyme leaves, raise the heat to medium high, and cook, breaking up the large pieces, until no longer pink, about 5 minutes.
Stir in the pasta sauce and dried cranberries, and season with salt and pepper to taste. Stir until warmed through, about 1 minute.
Remove the cooked squash from the oven. Fill each squash "bowl" evenly with the turkey mixture. Top each with additional thyme leaves as desired.