Slice the leeks into 1/2-inch rounds, making sure to use only the white and light green parts—this gives you the sweetest flavor without bitterness. Peel and cut the potatoes into 1-inch chunks so they cook evenly. Tear the kale into bite-sized pieces and remove all stems. Mince the fresh dill and set aside. Having everything prepped and ready will let you move through the cooking process smoothly without interruption.

Melt the butter in a large pot over medium heat until it's foaming. Add the sliced leeks and cook for 3-4 minutes, stirring occasionally, until they start to soften and become fragrant—this gentle cooking develops their natural sweetness. Add the potato chunks and cook for another 2-3 minutes, stirring to coat everything in the butter. This foundational step builds flavor depth that will carry through the entire soup.

Pour in the vegetable broth and bring the mixture to a boil over medium-high heat. Once boiling, reduce the heat to medium-low and simmer for 12-15 minutes until the potatoes are completely tender and break apart easily when pierced with a fork. The potatoes need to be fully cooked to create that creamy, velvety texture when blended.

Stir in the kale pieces, minced dill, salt, and pepper. Simmer for 2-3 minutes just until the kale is tender and wilts into the soup. I like to add the dill at this stage rather than earlier because it preserves the fresh, aromatic quality—cooking it too long dulls its bright flavor. The kale adds nutrition and a subtle earthiness that balances the richness of the cream.

Remove the pot from heat and let the soup cool for 2-3 minutes so it's safe to blend. Using an immersion blender, blend the soup in batches or directly in the pot until completely smooth and creamy. If using a countertop blender, work in batches and return the blended soup to the pot. The goal is a velvety texture with no chunks remaining.

Return the pot to medium heat and stir in the milk slowly, stirring constantly to incorporate it smoothly. Once the milk is fully mixed in and the soup is warm, remove from heat and stir in the cheddar cheese and dry mustard until the cheese melts completely. The dry mustard adds a subtle depth that enhances the cheese flavor without making the soup taste mustard-y. Let the soup sit off heat for 1-2 minutes so the cheese fully integrates.

Ladle the creamy soup into bowls and serve immediately while it's hot. This soup is naturally comforting and pairs beautifully with crusty bread for dipping.