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Remove edamame from plants and wash pods. Bring a small pot of salted water to boil and throw in beans, cooking for about 5 minutes or until tender but not mushy. If you can, save the water for boiling the pasta or bring a new pot to the boil.
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Halve the leeks lengthwise then thinly slice them crosswise. Wash them well. Cook the leeks in the oil and butter with 1 tsp salt and 1/2 tsp pepper in a large heavy skillet over medium heat until the leeks are golden, about 6 minutes.
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Add the edamame and cream and boil until the liquid is slightly thickened, about 4 minutes.
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Boil the pasta in the bean water until al dente. Drain. Toss the pasta with the leek, edamame mixture, Parmesan and season with salt and pepper to taste.