Preheat the oven to 400°F (200°C).
Cut the top off the bulb of garlic to expose the cloves and drizzle with a little olive oil. Wrap the bulb in aluminum foil and roast in the oven for about 30 minutes until soft and caramelized.
While the garlic is roasting, heat 2 tablespoons of olive oil in a medium pot over medium heat. Add the diced onion and sauté until translucent, about 5 minutes.
Add the cauliflower florets to the pot and stir to combine. Cook for about 5 minutes.
Pour in the vegetable broth and bring to a boil. Reduce heat and simmer for 15 minutes, or until the cauliflower is tender.
Once the garlic is done, let it cool slightly, then squeeze the roasted garlic cloves out of their skins into the pot with the cauliflower.
Using a blender or immersion blender, puree the soup until smooth.
Stir in the milk, shredded cheddar cheese, salt, pepper, and smoked paprika (if using). Heat gently until the cheese is melted and the soup is warmed through.
Serve hot, garnished with fresh chives or parsley if desired.