Sauté the aromatics. Warm the olive oil or butter in a large pot over medium heat. Add the chopped onion and a pinch of salt.
Cook 5–7 minutes until soft and translucent. Stir in the garlic and cook 30 seconds until fragrant.
Bloom the spices. Add cumin, coriander, and nutmeg. Stir for 30–45 seconds. Blooming spices in fat wakes up their flavor and gives the soup depth.
Add pumpkin and broth. Stir in the pumpkin puree, then pour in the broth.
Mix well, scraping up any browned bits from the pot. Bring to a gentle simmer.
Simmer for flavor. Reduce heat to low and simmer uncovered for 12–15 minutes. This helps the flavors meld and the soup thicken slightly.
Blend until smooth. Use an immersion blender directly in the pot to puree until silky.
Or carefully transfer to a blender in batches. Vent the blender lid and cover with a towel to avoid steam build-up.
Finish with cream and balance. Return the soup to low heat (if blended in a separate blender). Stir in cream or coconut milk. Add maple syrup or honey to round out the flavor, and vinegar or lemon juice to brighten.
Taste and adjust salt and pepper.
Serve and garnish. Ladle into bowls and add a swirl of cream or yogurt, a sprinkle of pepitas, chopped chives, or a drizzle of chili oil for heat.