Preheat the oven to 400°F (200°C).
On a baking sheet, toss the garlic cloves with 1 tablespoon of olive oil. Wrap them in aluminum foil and roast in the preheated oven for about 20 minutes or until they become soft and fragrant.
While the garlic is roasting, heat the remaining tablespoon of olive oil in a large pot over medium heat. Add the chopped onion and sauté for 5 minutes until translucent.
Add the chopped cauliflower, vegetable broth, salt, pepper, and thyme to the pot. Bring to a boil, then reduce the heat and let it simmer for 15 minutes, or until the cauliflower is tender.
Once the garlic is done roasting, carefully unwrap the foil and squeeze the roasted garlic out of its skins into the pot with the cauliflower.
Using a blender or an immersion blender, puree the soup until smooth and creamy.
Stir in the milk and shredded cheddar cheese, allowing the cheese to melt completely. Adjust seasoning if needed.
Serve the soup hot, garnished with fresh chives if desired.