Peel and dice the potatoes into uniform ½-inch cubes, finely dice the onion, mince the garlic, and dice the bacon if using. Shred the cheddar cheese from a block and let it sit at room temperature, then chop the fresh chives and parsley and set aside.
Add the diced bacon to a cold Dutch oven, then turn the heat to medium and cook for 6 to 8 minutes, stirring occasionally, until crispy. Remove the bacon with a slotted spoon to a paper towel-lined plate and leave about 1 tablespoon of rendered fat in the pot.
Add the butter to the pot with the bacon fat and melt over medium heat, then add the diced onion and cook for 5 minutes until softened. Add the minced garlic, dried thyme, rosemary, and smoked paprika and stir for 1 to 2 minutes until fragrant.
Sprinkle the flour over the onion and garlic mixture and stir continuously for 2 minutes until the flour evenly coats the vegetables and the raw flour smell disappears. This step is essential for thickening the soup.
Slowly pour in the chicken broth one cup at a time, stirring constantly to prevent lumps, then add the diced potatoes and stir to combine. Bring to a boil over medium-high heat, then reduce to medium-low, cover, and simmer for 15 to 20 minutes until the potatoes are fork-tender.
Reduce the heat to low and pour in the whole milk and heavy cream. Stir gently and allow the soup to warm for 3 to 4 minutes without boiling.
For a partially creamy texture, use an immersion blender to blend just the bottom of the pot for 10 to 15 seconds, leaving plenty of chunks. For a fully smooth soup, blend completely; for a chunky soup, skip this step entirely.
With the heat on the lowest setting, add the shredded cheddar cheese in two or three batches, stirring after each addition until fully melted and smooth. Stir in the fresh chives and parsley, then season generously with salt and black pepper to taste.
Ladle the soup into bowls and top each with extra shredded cheddar, reserved crispy bacon, a dollop of sour cream, chopped chives, and a dusting of smoked paprika. Serve immediately with crusty bread or dinner rolls.