Melt the butter in a large, deep stockpot or Dutch over medium-high heat. Add the onion and carrots and sauté until just beginning to soften; about 5 minutes. Add the jalapeno, garlic, salt, and cayenne pepper and cook the garlic is fragrant; about 30 seconds.
Sprinkle the flour over the top of the vegetables, then cook for an additional minute until lightly golden and no white bits remain. Slowly stir in the milk and cook until the mixture thickens, about 2 minutes, stirring constantly.
Stir in the broth, then add the chopped potato. Bring to boil, then partially cover and let simmer until the vegetables are tender, about 15 to 20 minutes. Stir from time to time and adjust the heat as needed so that you maintain a steady simmer (but not a full-on boil).
Transfer the soup to a blender or food processor and puree in batches until smooth (alternatively, you can use an immersion blender directly in the pot).
Stir in the cheese a half-cup at a time, allowing it to melt completely before stirring in the next half cup. Serve garnished with chives, bacon, and additional cheddar cheese.
Notes
TO MAKE IN A SLOW COOKER: After Step 2, transfer the vegetable mixture to a 5-quart or larger crockpot. Add the chicken stock and potato. Cover and cook until the vegetables are tender, about 5 to 6 hours on low. Proceed with the recipe as directed.
TO STORE: Leftover potato cheese soup may be refrigerated for up to 5 days in an airtight container.
TO REHEAT: Warm gently on the stovetop or in the microwave (at a reduced power level) until steaming. Do not warm too quickly or the soup may separate.
TO FREEZE: Potato cheese soup can be frozen for up to 3 months. For best results, allow the soup to cool completely before putting it in the freezer and defrost overnight in the fridge before warming. Note that dairy-based soups do tend to curdle and separate when frozen and thawed. If this bothers you, then I would not recommend freezing this soup.