01 - Melt the butter in a large pot over medium heat. Add the chopped onion and cook for 3-4 minutes until softened and translucent.
02 - Stir in the minced garlic, diced celery, and carrot. Continue cooking for another 3 minutes until fragrant and vegetables begin to soften.
03 - Add the diced potatoes to the pot. Season with sea salt, black pepper, and smoked paprika if using. Toss to coat evenly.
04 - Pour in the broth and bring to a boil. Reduce heat to low 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 purée the soup, leaving some potato chunks for texture. Alternatively, blend half the soup in a countertop blender and return to the pot.
06 - Stir in the whole milk and heavy cream. Return to low heat and warm gently, being careful not to boil the mixture.
07 - Add the grated Irish cheddar in batches, stirring constantly until fully melted and incorporated into a smooth, creamy consistency.
08 - Stir in the chopped fresh chives. Taste and adjust seasoning with additional salt and pepper if needed.
09 - Ladle the hot soup into serving bowls. Garnish with additional fresh chives and extra shredded cheddar cheese if desired. Serve immediately with crusty bread.