01 - Melt butter in a large pot over medium heat. Add chopped onion and cook for 3–4 minutes until softened and translucent.
02 - Stir in minced garlic, diced celery, and carrot. Continue cooking for 3 minutes until fragrant and vegetables begin to soften.
03 - Add diced potatoes to the pot. Season with sea salt, black pepper, and smoked paprika, tossing to coat evenly.
04 - Pour in broth and bring to a boil. Reduce heat to low and simmer uncovered for 20–25 minutes until potatoes are completely tender and easily pierced with a fork.
05 - Remove from heat. Use an immersion blender to partially purée the soup, leaving some potato chunks for texture. Alternatively, blend half the soup in a countertop blender and return to pot.
06 - Stir in whole milk and heavy cream. Return to low heat and warm gently, being careful not to boil the mixture.
07 - Add grated Irish cheddar in batches, stirring constantly until fully melted and smooth. Ensure each addition is incorporated before adding more cheese.
08 - Stir in chopped fresh chives. Taste and adjust seasoning with additional salt and pepper if needed.
09 - Ladle hot soup into bowls. Garnish with additional fresh chives and extra grated cheddar if desired. Serve immediately with crusty bread.