01 - In a large pot, melt the butter over medium heat. Add the onion and sauté for 3–4 minutes until softened and translucent.
02 - Stir in the garlic, celery, and carrot. Cook for another 3 minutes until fragrant and vegetables begin to soften.
03 - Add the diced potatoes and season with salt, pepper, and smoked paprika if using. Stir to coat evenly.
04 - Pour in the broth, bring to a boil, then reduce heat and simmer uncovered for 20–25 minutes until potatoes are very tender and easily pierced with a fork.
05 - Remove from heat. Use an immersion blender to partially blend the soup, leaving some chunks for texture. Alternatively, blend half the soup in a countertop blender and return to the pot.
06 - Stir in the milk and heavy cream. Return to low heat and warm gently, being careful not to boil.
07 - Add the grated cheddar in batches, stirring constantly until completely melted and smooth.
08 - Stir in the chopped chives. Taste and adjust seasoning with additional salt and pepper if needed.
09 - Ladle into bowls, garnish with extra chives and additional cheddar if desired. Serve hot with crusty bread.