Add the potatoes, onion, garlic, celery, carrots, thyme, bay leaf, smoked paprika, butter, salt, and pepper to the crockpot.
Pour in the broth and stir to combine.
Cover and cook on Low for 6–8 hours or High for 3–4 hours, until the potatoes are very tender and easily mash with a spoon.
Remove the bay leaf. Using an immersion blender, blend the soup in the crockpot until it’s mostly smooth, leaving a few chunks if you like texture. If you don’t have an immersion blender, carefully transfer half the soup to a blender, puree, and return it to the crockpot.
Stir in the cream or half-and-half and the cream cheese until fully melted and smooth.
If needed, cover and cook on Low for another 10–15 minutes to warm through.
Add the shredded cheddar a handful at a time, stirring until melted. Taste and adjust seasoning with more salt and pepper.
Serve hot with your favorite toppings: crumbled bacon, extra cheddar, green onions or chives, and a dollop of sour cream.