In a large pot, melt the butter (or heat olive oil and butter) over medium heat. Add the finely chopped onion and cook until translucent and soft, about 5-7 minutes. Add the minced garlic and cook for another 1-2 minutes until fragrant. Sprinkle the flour over the onion and garlic mixture, stirring constantly to form a roux; cook for 2 minutes to eliminate the raw flour taste. Gradually whisk in the chicken or vegetable broth, ensuring no lumps form. Add the cubed potatoes to the pot and bring the soup to a boil. Reduce heat and simmer uncovered for 20-25 minutes until potatoes are tender. Using an immersion blender, blend part of the soup to achieve a creamy texture while leaving some potato chunks for texture. Alternatively, remove about half the soup, blend in a blender, and return to pot. Stir in the whole milk and heavy cream, warming gently but do not boil. Add shredded sharp cheddar cheese, stirring until fully melted and incorporated. Mix in fresh parsley, chives, thyme, salt, black pepper, and optional cayenne or smoked paprika. Adjust seasoning to taste. Serve hot, garnished with your choice of crumbled bacon, extra cheddar, sour cream, fresh herbs, or croutons.