Preheat your oven to 400°F (200°C).
Slice the top off the head of garlic to expose the individual cloves. Drizzle with 1 tablespoon of olive oil and wrap it in aluminum foil. Roast in the oven for about 30 minutes, or until the cloves are soft and golden.
While the garlic roasts, cut the cauliflower into florets. In a large pot, heat the remaining tablespoon of olive oil over medium heat. Add the chopped onion, cooking until softened and translucent, about 5 minutes.
Add the cauliflower florets to the pot with the onion and stir for another 3 minutes.
Pour in the vegetable broth and bring the mixture to a boil. Reduce the heat to low and simmer for about 15 minutes, until the cauliflower is tender.
Once the garlic is roasted, squeeze the soft cloves out of their skins and add them to the soup pot.
Use an immersion blender to puree the soup until smooth. If using a regular blender, allow the soup to cool slightly before blending in batches, then return to the pot.
Stir in the whole milk and shredded cheddar cheese, continuing to heat over low until the cheese is melted and incorporated. Season with salt and pepper to taste.
Serve the soup hot, garnished with fresh chives or parsley if desired.