Preheat the oven to 425°F and line a large baking sheet with parchment paper.
Toss the cauliflower florets with olive oil, salt, and pepper on the prepared baking sheet until evenly coated and spread them into a single layer.
Roast the cauliflower for 20 to 25 minutes, stirring once halfway through, until the edges are golden and the florets are tender and lightly caramelized.
While the cauliflower roasts, melt the butter in a large soup pot over medium heat.
Add the chopped onion to the pot and cook, stirring occasionally, for 4 to 5 minutes until softened and translucent.
Stir in the minced garlic and cook for about 30 seconds until fragrant, being careful not to let it brown.
Add the roasted cauliflower to the pot along with the chicken broth, smoked paprika, and thyme, then stir to combine everything well.
Bring the mixture just to a gentle boil over medium heat, then reduce the heat to low and simmer for 8 to 10 minutes to allow the flavors to meld.
Remove the pot from the heat and use an immersion blender to blend the soup until completely smooth and velvety, adjusting the angle of the blender to catch any larger pieces of cauliflower.
Return the pot to low heat and stir in the heavy cream until fully incorporated and the soup is gently steaming but not boiling.
Add the shredded cheddar and Parmesan cheeses a small handful at a time, stirring constantly after each addition until melted and the soup is thick and creamy.
Taste the soup and season with additional salt and pepper as needed, adjusting the thickness with a splash of extra broth if you prefer a thinner consistency.
Ladle the soup into bowls and top with bacon crumbles, extra shredded cheddar, chopped chives, and a pinch of smoked paprika if using, then serve hot.
To store leftovers, let the soup cool completely, transfer to airtight containers, and refrigerate for up to three days or freeze for up to two months.
Reheat the soup gently on the stovetop over low heat, stirring often and adding a splash of broth or cream if needed to restore a silky texture.