Preheat your oven to 400°F (200°C). Brush the cut sides of the squash with olive oil and season with salt and pepper.
Place the squash cut-side down on a baking sheet and roast for 25-30 minutes, until fork-tender but still firm enough to hold the filling.
While the squash roasts, heat olive oil in a large skillet over medium heat. Add onions and cook until translucent, about 5 minutes.
Add mushrooms and garlic to the skillet. Cook until mushrooms release their moisture and begin to brown, approximately 8-10 minutes.
Stir in the spinach and cook until wilted, about 2-3 minutes. Remove from heat and drain any excess liquid.
Mix in the cooked quinoa or rice, Parmesan cheese, thyme, salt, and pepper.
Flip the roasted squash cut-side up and fill each half generously with the mushroom-spinach mixture.
Top with additional Parmesan cheese and return to the oven for 10-15 minutes, until the cheese is melted and golden.