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Preheat oven to 400 degrees F.
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In baking dish, set squash, cut-side up and brush with olive oil, coating entire surface of flesh. Season with salt and pepper. Roast 35-40 minutes, or until flesh is tender when pierced with fork. Once done, remove from oven and let cool slightly.
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In large skillet over medium heat, heat 1 tablespoon oil, add onion and garlic, and sauté 3-4 minutes until softened.
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To skillet, add beef and cook until browned, about 5-7 minutes, breaking it up as it cooks.
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Stir in broccoli, thyme, oregano and season salt. Cook 4-5 minutes, until broccoli is tender.
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Once squash is cool enough to handle, scoop about 1/2-cup flesh from each squash half and add it to beef and broccoli mixture in skillet. Mix well to combine.
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Stuff squash generously with beef and broccoli mixture.
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Sprinkle grated Parmesan, dairy-free cheese or nutritional yeast, if using, on top of each squash.
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Return squash to oven to bake additional 10-15 minutes, until tops are golden and everything is heated through.
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Remove from oven and garnish with fresh parsley.