In your large pot or Dutch oven, cook the bacon over medium heat until crispy. Remove bacon to a paper towel-lined plate to drain, reserving about 1-2 tablespoons of bacon grease in the pot if desired (drain excess). Once cool, crumble the bacon.

Add the butter (and reserved bacon grease, if using) to the pot over medium heat. Once melted, add the chopped onion and celery. Cook, stirring occasionally, until the vegetables are softened and translucent, about 5-7 minutes. Add the minced garlic and cook for another minute until fragrant, being careful not to burn it.

Sprinkle the flour over the sautéed vegetables. Stir constantly with a whisk or wooden spoon for 1-2 minutes. This cooks the raw taste out of the flour. The mixture will look like a paste.

Slowly pour in the chicken or vegetable broth, whisking continuously as you pour to prevent lumps. Bring the mixture to a gentle simmer, scraping the bottom of the pot to ensure no flour mixture is sticking.

Add the cubed potatoes to the pot. Stir to combine. Bring the soup back to a gentle simmer.

Reduce heat to low, cover, and let the soup simmer for 15-20 minutes, or until the potato cubes are very tender when pierced with a fork.

Remove the pot from the heat. Use a potato masher to mash some of the potatoes against the side of the pot. Mash more for a smoother soup, less for a chunkier one. Alternatively, use an immersion blender to blend some or all of the soup to your desired consistency. Stir in the milk or half-and-half and about 1 cup of the shredded cheddar cheese.

Return the pot to low heat. Stir gently until the cheese is melted and the soup is heated through. Do not boil the soup after adding the dairy, as this can cause it to curdle. Stir in the sour cream. Season with salt and pepper to taste. Add a pinch of cayenne or smoked paprika now if desired.

Ladle the hot soup into bowls. Garnish generously with the remaining shredded cheddar cheese, crumbled bacon, and fresh chives or a dollop of sour cream.