Remove the leaves and thick center stalk from the cauliflower. Cut the cauliflower into similarly sized florets and rinse them thoroughly.
Heat the butter and olive oil in a large pot or Dutch oven over medium-low heat.
Add the diced onion and cook for 5–7 minutes, stirring occasionally, until soft and translucent. Do not let the onion brown.
Add the garlic, turmeric, and nutmeg. Cook for approximately 30 seconds, stirring constantly.
Add the cauliflower florets and pour in 2½ cups of broth. Season with salt and black pepper.
Bring the soup to a gentle boil. Reduce the heat, cover the pot, and simmer for 18–20 minutes, or until the cauliflower is very tender.
Remove the pot from the heat. Carefully blend the soup with an immersion blender until completely smooth. Keep the blender head submerged to prevent splashing.
Return the pot to low heat. Add the finely grated Parmesan a little at a time, stirring continuously until it melts smoothly into the soup.
Do not allow the soup to boil after adding the Parmesan.
Taste and adjust the seasoning. If the soup is too thick, add warm broth about ¼ cup at a time until it reaches your preferred consistency.
Divide the soup among bowls. Garnish with shaved Parmesan, olive oil, fresh herbs, black pepper, or red pepper flakes.