Preheat oven to 425°. Divide cauliflower between two large baking sheets. Drizzle each baking sheet with a tablespoon of olive oil and season with salt and pepper. Toss to combine and scatter 4 thyme sprigs on top.
Bake until cauliflower is golden and tender, about 20 minutes.
In a large pot (or dutch oven) over medium heat, cook bacon until crispy, about 8 minutes. Drain cooked bacon slices on a paper towel-lined plate, reserving fat in the pot, then chop bacon into small pieces.
Add onion to bacon fat and cook until tender, about 5 minutes. Stir in garlic and cook until fragrant, another minute, then pour in wine. Cook until wine has mostly reduced down, about 2 minutes.
Add roasted cauliflower to the pot and pour over chicken broth. Add bay leaf and remaining thyme sprigs. Bring mixture to a boil, reduce heat, and simmer until the cauliflower is falling apart, about 20 minutes.
Remove bay leaf and thyme sprigs from soup. Using an immersion blender, purée soup until smooth.
Season to taste with salt and pepper then stir in heavy cream. Serve warm with cheddar, chopped bacon, and chives.