Preheat the oven to 400°F. Line one or two baking sheets with parchment paper.
Wash and dry the sugar pumpkins. Carefully cut off the tops around the stems, or cut the pumpkins in half from stem to base. Scoop out the seeds and fibrous pulp.
Brush the pumpkin cavities and cut surfaces with 2 tablespoons olive oil. Season with ½ teaspoon salt and ¼ teaspoon black pepper.
If using pumpkin halves, place them cut-side down on the prepared baking sheets. Roast for 25 to 35 minutes, until the flesh is mostly tender when pierced with a fork but the pumpkins still hold their shape.
While the pumpkins roast, heat 1 tablespoon olive oil in a large skillet over medium heat. Add the sausage and cook for 6 to 8 minutes, breaking it into crumbles, until browned and fully cooked. Transfer the sausage to a bowl.
Add the onion and mushrooms to the skillet. Cook for 5 to 6 minutes, until softened and lightly browned. Add the garlic and cook for 30 seconds.
Stir in the cooked wild rice, spinach, dried cranberries, pecans, sage, thyme, broth, cooked sausage, salt, and black pepper. Cook for 2 to 3 minutes, until the spinach wilts and the filling is heated through.
Remove the pumpkins from the oven and carefully turn the halves cut-side up if needed. Spoon the filling into each pumpkin cavity.
Top with goat cheese or Parmesan. Return the stuffed pumpkins to the oven and bake for 10 to 15 minutes, until the filling is hot and the cheese is softened or melted.
Garnish with parsley, if desired, and serve warm.