Preheat the oven to 425°F. Toss the cauliflower florets with the olive oil and 1/2 teaspoon kosher salt on a rimmed baking sheet. Spread into a single layer and roast until edges are lightly browned and tender, 18–22 minutes. Roasting adds a deeper, 'loaded' flavor.
While the cauliflower roasts, cook the bacon in a skillet over medium heat until crisp. Transfer to a paper towel-lined plate to drain, then crumble. Leave 1 tablespoon bacon fat in the pan (or use 1 tablespoon butter) for the next step.
Reduce heat to medium and add the butter to the skillet with the bacon fat. Sauté the diced onion until soft and translucent, about 5–6 minutes, then stir in the minced garlic and smoked paprika and cook 30 seconds until fragrant.
Transfer the roasted cauliflower to a large pot, add the sautéed onion and garlic, then pour in the chicken broth. Bring to a gentle simmer over medium heat and cook uncovered for 10 minutes so the flavors meld.
Use an immersion blender to purée the soup until mostly smooth but still slightly textured for body. (If you don't have an immersion blender, carefully puree the soup in batches in a stand blender.)
Return the pot to low heat and stir in the heavy cream and sour cream until fully incorporated. Add the shredded cheddar a handful at a time, stirring after each addition so it melts evenly. Taste and adjust seasoning with the remaining 1 teaspoon kosher salt and black pepper.
Warm the soup gently for 2–3 minutes more—do not boil after adding the dairy to prevent separation. If the soup is thicker than you like, thin it with up to 1/2 cup extra broth or water.
Ladle the soup into bowls and finish each with crumbled bacon, a sprinkle of extra cheddar if using, and a generous scattering of sliced chives.
Let leftovers cool to room temperature, then refrigerate in an airtight container for up to 3 days. Reheat gently on the stovetop, stirring so the soup stays creamy.