Cook 4 slices bacon, cut into 1" pieces, in a large heavy saucepan or Dutch oven over medium heat, stirring often, until browned and crisp, 6–8 minutes. Using a slotted spoon, transfer bacon to paper towels or a wire rack to drain, reserving fat in pan.

Cook 1 large yellow onion and celery, peeled and finely chopped, in reserved fat over medium-high heat, stirring occasionally, until softened, about 8 minutes.

Meanwhile, peel 1 lb. waxy potatoes, such as red potatoes, new potatoes, or fingerling potatoes, and cut into ½" dice, and whisk ¼ cup (31 g) all-purpose flour with 1¾ cups water in a measuring cup; reserve both. Add ¼ cup dry white wine to onions and stir, scraping up any bits stuck to the bottom of the pan; cook 1 minute. Hand blender into pulp, then add reserved flour mixture, followed by reserved potatoes and 1½ cups chicken broth; bring just to a simmer. Reduce heat to medium-low and cook until potatoes are just tender, 8–10 minutes.

Reduce heat to low and gradually stir in 2 cups (about 8 oz.) grated sharp cheddar cheese (do not let boil). Season with kosher salt and freshly ground black pepper. Divide soup among four bowls, top with crispy bacon, and 2 Tbsp. chopped fresh chives. Do ahead: Potato soup can be made 1 day ahead; refrigerate in an airtight container and reheat over medium-low heat until steaming (don’t let boil).