Caramelize the Onions: In a large skillet, heat the olive oil and butter over medium heat. Add the sliced onions and a pinch of salt. Cook for about 15-20 minutes, stirring occasionally until the onions are golden brown and caramelized. If they start to stick, lower the heat and add a tablespoon of the broth to deglaze the pan.
Add Pumpkin: Once the onions are caramelized, add the diced pumpkin to the skillet, along with the brown sugar and thyme. Stir to combine, ensuring the pumpkin is coated with the flavorful onion mixture.
Season and Cook: Add the remaining vegetable broth (or water), nutmeg, salt, and pepper to the skillet. Stir well, then bring the mixture to a gentle simmer. Cover the skillet and cook for about 10-15 minutes, or until the pumpkin is tender, stirring occasionally.
Finishing Touch: If using Parmesan cheese, sprinkle it over the pumpkin mixture in the final few minutes of cooking, allowing it to melt slightly. Adjust seasoning to taste.
Serve: Remove from heat and garnish with chopped fresh parsley before serving. This dish pairs beautifully with crusty bread or can be served over rice or quinoa for a complete meal.