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 the remaining butter in a skillet over medium heat, and saute the mushrooms 5 minutes, until lightly browned. Stir into soup.
In a small, mix the half-and-half and flour until smooth. Stir into the soup to thicken. Garnish each bowl of soup with fresh dill to serve.
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