Preheat the oven to 400°F (200°C).
On a baking sheet, toss the cauliflower florets with olive oil, salt, and pepper. Wrap the whole head of garlic in aluminum foil without cutting it. Place it on the baking sheet alongside the cauliflower.
Roast the cauliflower and garlic in the preheated oven for about 25 minutes, or until the cauliflower is tender and slightly browned.
In a large pot, sauté the chopped onion over medium heat until translucent (about 5 minutes).
Once the cauliflower and garlic are roasted, add the cauliflower florets to the pot with the onions. Squeeze out the roasted garlic cloves from their skins and add them into the pot as well.
Pour in the vegetable or chicken broth, and bring the mixture to a simmer. Let it cook for about 10 minutes.
Remove the pot from heat and use a blender or immersion blender to puree the soup until smooth.
Return the pureed soup to the pot, and stir in the shredded cheddar cheese and heavy cream. Heat over low, stirring until the cheese is melted and the soup is heated through.
Adjust seasoning with salt and pepper as needed, and serve hot, garnished with fresh chives if desired.