Start with the aromatics. Place a large pot over medium heat and melt 3 tablespoons of butter. Add the chopped leeks and sliced carrots. Cook for about 5 minutes, stirring occasionally, until the vegetables begin to soften and release their aroma.

Build the broth. Pour in the chicken broth and stir in the dried dill, salt, pepper, and bay leaf. Add the diced potatoes and bring the mixture to a gentle simmer. Cover the pot and let it cook for about 20 minutes, or until the potatoes are tender but still hold their shape. Remove the bay leaf once cooking is complete.

Cook the mushrooms separately. In a skillet, melt the remaining butter over medium heat. Add the sliced mushrooms in a single layer and sauté for about 5 minutes, allowing them to brown lightly and release their juices. Once done, transfer the mushrooms into the soup pot and stir to combine.

Create a creamy finish. In a small bowl, whisk together the half-and-half and flour until smooth and lump-free. Slowly pour this mixture into the soup while stirring continuously. Let the soup simmer for a few more minutes until it thickens slightly and becomes creamy.

Serve and garnish. Ladle the hot soup into bowls and finish with a sprinkle of fresh dill on top for a bright, herbal touch.