Preheat your oven to 400°F (200°C).
Prepare the garlic by slicing the top off the whole head, drizzling with 1 tablespoon of olive oil, and wrapping it in aluminum foil. Place it on a baking sheet.
In a large bowl, toss the cauliflower florets with 1 tablespoon of olive oil, salt, and pepper. Spread the cauliflower on the baking sheet next to the garlic.
Roast the cauliflower and the wrapped garlic in the preheated oven for about 25-30 minutes, or until the cauliflower is golden-brown and tender, and the garlic cloves are soft. Remove from the oven and let cool slightly.
In a large pot over medium heat, add the diced onion. Sauté for about 5 minutes, or until the onion is soft and translucent.
Squeeze the roasted garlic cloves out of their skins into the pot with the onion. Add the roasted cauliflower and vegetable broth. Stir to combine and bring to a simmer for about 5 minutes.
Using a blender or immersion blender, carefully purée the soup until smooth. If using a blender, you may need to do this in batches.
Return the pureed soup to the pot. Stir in the heavy cream and shredded cheddar cheese until the cheese is melted and the soup is heated through. Season with additional salt and pepper to taste.
Serve hot, garnished with fresh chives if desired.