Cook the Cauliflower: In a large pot, bring the chicken or vegetable broth to a boil. Add the cauliflower florets and cook until tender, about 10-15 minutes.
Sauté the Aromatics: While the cauliflower is cooking, heat the olive oil or butter in a skillet over medium heat. Add the chopped onion and cook until translucent, about 5 minutes. Add the minced garlic and cook for another 1-2 minutes until fragrant.
Blend the Soup: Once the cauliflower is tender, use an immersion blender to blend the cauliflower and broth until smooth. Alternatively, transfer the cauliflower and broth in batches to a blender and blend until smooth.
Combine Ingredients: Return the blended cauliflower to the pot. Stir in the sautéed onions and garlic, heavy cream, dried thyme, and dried rosemary. Heat over medium heat until warmed through.
Add Cheese and Bacon: Stir in the shredded cheddar cheese until melted and smooth. Season with salt and pepper to taste.
Serve: Ladle the soup into bowls. Garnish with crumbled bacon, extra shredded cheddar cheese, and chopped green onions.