Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente. Drain, reserving 1 cup of the pasta cooking water.
In a large, deep skillet, heat the oil, 4 turns of the pan, over medium-high heat. Add the shiitakes and cook until browned, about 7 minutes. Add the leeks and wilt for 2 minutes, then stir in the ginger and garlic for 1 minute. Stir in the edamame to heat through. Stir in the sake and season liberally with salt and pepper.
Stir the pasta and reserved pasta cooking water into the skillet, then add the sesame seeds and sesame oil. Turn off the heat and toss for 1 full minute.