Melt 3 tablespoons butter in a large saucepan over medium heat. Mix in leeks and carrots, and cook 5 minutes. Pour in broth. Season with dill, salt, pepper, and bay leaf. Mix in potatoes, cover, and cook 20 minutes, or until potatoes are tender but firm. Remove and discard the bay leaf.

Melt remaining butter in a skillet over medium heat. Sauté mushrooms until lightly browned, about 5 minutes. Stir mushrooms into the soup.

Mix half-and-half and flour in a small bowl until smooth. Stir mixture into the soup to thicken. Garnish each bowl with fresh dill before serving.