In a large saucepan or Dutch oven over medium heat, cook the bacon, stirring frequently, until crisp, 8 to 10 minutes. Using a slotted spoon, transfer the bacon to a paper towel-lined plate and set aside.
Return the pot to the stove over medium heat. Add the onion and garlic and cook, stirring frequently, until the onion is soft and translucent, about 5 minutes. Do not brown; reduce the heat if necessary. Add the flour and stir until dissolved, about 30 seconds.
Pour in the chicken broth, stirring with a wooden spoon to scrape any brown bits off the bottom of the pan. Increase the heat to high and bring to a boil. Carefully add the potatoes, along with the salt, pepper, and thyme, and bring to a boil again. Reduce the heat to low, cover, and cook until the potatoes are very tender, 20 to 25 minutes.
Off the heat, use an immersion blender to purée the soup until completely smooth. (Alternatively, cool the soup slightly, then puree in a blender in batches. Be sure to leave the center hole in the lid, and cover with a kitchen towel, to allow the steam to escape.) Add the sour cream and butter and whisk by hand until the butter is melted and the sour cream is incorporated. Taste and adjust the seasoning if necessary. Ladle the soup into bowls and top with the chives, cheese, and reserved bacon.