Preheat your oven to 400°F (200°C). Wrap the whole garlic cloves in aluminum foil and place them on a baking sheet. Roast for 30 minutes until soft and caramelized.
In a large pot, heat the olive oil over medium heat. Add the diced onion and sauté until translucent, about 5 minutes.
Add the chopped cauliflower to the pot and stir for another 5 minutes until slightly softened.
Pour in the vegetable or chicken broth, along with salt, black pepper, and paprika. Bring to a boil, then reduce heat and let it simmer for about 15 minutes, until the cauliflower is tender.
Once the garlic is roasted, squeeze the cloves out of their skins and add them to the pot.
Use a blender or immersion blender to puree the soup until smooth. If using a regular blender, blend in batches, being cautious of the hot liquid.
Return the pureed soup to the pot over low heat. Stir in the heavy cream and shredded cheddar cheese, allowing the cheese to melt completely and the soup to heat through.
Taste and adjust seasoning if needed. Serve hot, garnished with fresh chives or parsley if desired.