Crisp the bacon: Add the chopped bacon to a large pot or Dutch oven.
Cook over medium heat, stirring occasionally, until browned and crisp, about 7–9 minutes. Use a slotted spoon to transfer bacon to a paper towel–lined plate. Leave 1–2 tablespoons of bacon fat in the pot.
If there’s less, add a bit of olive oil or butter.
Sauté aromatics: Add the onion and a pinch of salt to the pot. Cook until translucent and lightly golden, 4–5 minutes. Stir in the garlic and cook 30 seconds, just until fragrant.
Add veggies and seasoning: Stir in the cauliflower and carrot.
Sprinkle in thyme, smoked paprika, and a few grinds of black pepper. Cook 2–3 minutes to bloom the spices and start softening the veggies.
Simmer: Pour in the chicken broth and add the bay leaf. Bring to a boil, then reduce to a gentle simmer.
Cover and cook until the cauliflower is very tender, 12–15 minutes.
Blend for creaminess: Remove the bay leaf. Use an immersion blender to blend about half to two-thirds of the soup right in the pot, leaving some chunks for texture. If using a countertop blender, carefully blend in batches and return the soup to the pot.
Add chicken and cream: Stir in the cooked chicken and half of the crispy bacon.
Pour in the heavy cream or half-and-half and simmer 3–5 minutes to warm through. If using Parmesan, stir it in now to melt.
Taste and adjust: Add salt and pepper to taste. If you want more richness, add another splash of cream.
For extra smoke, add a pinch more smoked paprika.
Serve: Ladle into bowls and top with the remaining bacon, fresh chives or parsley, and a pinch of red pepper flakes if you like heat. Serve with crusty bread or a simple green salad.