Prep the base: Peel and dice the potatoes into 1/2- to 3/4-inch cubes. Dice the onion, carrots, and celery.
Mince the garlic. This ensures even cooking and a creamy blend later.
Layer in the slow cooker: Add potatoes, onion, carrots, celery, garlic, thyme, bay leaf, and a generous pinch of salt and pepper. Pour in the broth.
Stir to combine.
Cook low and slow: Cover and cook on Low for 6–8 hours or High for 3–4 hours, until the potatoes are very tender and easily smashed with a spoon.
Make a quick roux: About 20 minutes before serving, melt the butter in a small pan. Whisk in the flour and cook for 1 minute. Slowly whisk in 1 cup of the hot soup liquid to make a smooth slurry.
Stir this back into the slow cooker. (Use 1 1/2 tablespoons cornstarch whisked with 2 tablespoons cold water if gluten-free.)
Mash or blend: Remove the bay leaf. Use a potato masher to mash some of the potatoes right in the slow cooker, leaving some chunks for texture. For extra silky soup, blend 2–3 cups in a blender or use an immersion blender, then return it to the pot.
Add the cream and cheese: Stir in the heavy cream and shredded cheddar until melted and smooth.
Taste and adjust seasoning with more salt and pepper as needed.
Crisp the bacon: If using bacon, cook until crisp, then crumble. Stir some into the soup and save some for topping.
Garnish and serve: Ladle into bowls and finish with bacon, chives, extra cheddar, and a small dollop of sour cream if you like.