In a large cast-iron pot over medium heat, add 1 tablespoon of butter. Add the bacon and cook until browned and the fat is rendered. Remove the bacon with a slotted spoon and place it on a paper towel-lined plate. Drain all the fat except for 2 tablespoons, keeping the pot unwashed. Add the diced carrot and onion, seasoning with salt and pepper. Sauté for 7–9 minutes until tender, without rushing. Then, add the gold potatoes, sweet potatoes, garlic, and seasonings, and sauté for an additional 2 minutes.
Pour in 4 cups of chicken broth and increase the heat to bring it to a boil. Once boiling, reduce heat to medium and simmer, uncovered, for 15 minutes or until potatoes are fork tender. Stir occasionally.
In a separate pot, melt the remaining 4 tablespoons butter over medium heat. Add flour and cook for 1 minute, whisking constantly. Gradually add milk while whisking. Continue to whisk until the mixture thickens (see note 5).
Reduce the heat to low and stir the milk mixture into the soup. If desired, use a potato masher to mash some of the potatoes while leaving others in chunks. Turn off the heat and gradually add the grated cheese, a handful at a time, stirring gently until melted. Taste and adjust the seasonings as needed.
Ladle soup into bowls, topping each with reserved bacon and optional sweet potato hash browns (see note 4). Garnish with fresh thyme if desired. Serve with crusty bread or rolls for dipping, or in bread bowls.
Notes
Note 1: Golden potatoes have thin skins, so no need to peel. If using a different variety, peel them first.Note 2: Low-fat milks don't offer the same thickness, creaminess, or flavor. It's best to stick with whole milk if possible.Note 3: Sharp Cheddar provides more flavor than mild or medium, which is crucial here. Use a block of cheese and shred it fresh for a smoother melt.Note 4:Sweet potato hash browns are the best topping for this soup! I pop them in the air fryer just before the soup is ready. You can also top this soup with fresh thyme and serve with hearty buttered bread.Note 5: Cook the mixture until it's thick enough to leave a clear trail on a wooden spoon. If you draw a line with your finger, it should stay put, not run together.Storage: Cool the soup to room temperature, then refrigerate in airtight containers for 3–5 days. For longer storage, freeze for 2–3 months. When reheating, do so on low heat, stirring frequently, and avoid boiling.