Preheat the oven to 400°F (200°C).
Peel the outer layers of the garlic bulb and slice off the top to expose the individual cloves. Wrap the garlic bulb in aluminum foil with a drizzle of olive oil. Bake in the preheated oven for about 30-35 minutes until soft and caramelized.
While the garlic is roasting, in a large pot, heat the remaining olive oil over medium heat. Add the diced onion and sauté for about 5 minutes until translucent.
Add the cauliflower florets to the pot and stir to combine. Cook for an additional 5 minutes, stirring occasionally.
Pour in the vegetable broth and bring the mixture to a boil. Reduce the heat and let it simmer for about 15 minutes, or until the cauliflower is tender.
Remove the garlic from the oven and let it cool slightly. Squeeze the roasted garlic cloves into the pot with the cauliflower.
Using an immersion blender, blend the soup until smooth. If using a countertop blender, carefully transfer the soup in batches and blend until creamy.
Return the blended soup to the pot over low heat. Stir in the shredded cheddar cheese and milk, mixing until the cheese is melted and incorporated. Season with salt and pepper to taste.
Serve hot in bowls, garnished with chopped chives or parsley if desired.