In a large pot, melt the butter or heat the olive oil over medium heat. Add the chopped onion and a pinch of salt. Cook, stirring occasionally, until the onion is soft and translucent, 6–8 minutes. Add the minced garlic and cook for 1 minute until fragrant.

Add the broccoli and cauliflower to the pot and stir to coat with the aromatics. Pour in the stock until the vegetables are mostly submerged. Bring to a gentle simmer, reduce heat, cover, and cook until the vegetables are tender, about 12–15 minutes.

Use an immersion blender to puree the soup until smooth. Alternatively, transfer the soup in batches to a countertop blender and blend until silky. Return the soup to low heat.

Stir in the milk or half-and-half and grated cheddar if using. Season with salt, pepper, and a pinch of nutmeg. Heat through gently without boiling. Serve with chopped chives, a drizzle of olive oil, and crusty bread.