Coat a large, straight-sided saute pan set over medium heat with the peanut oil and toss in the chicken and 1 tablespoon of the soy sauce. Cook until the chicken is cooked through, then remove and reserve for later use.
Add in the white scallion pieces, garlic, ginger and jalapenos. Add in a splash more of the oil and cook over medium heat, 1 to 2 minutes. Toss in the carrots and cook until softened, about 2 minutes. Toss in the mushrooms and cook until softened, 2 to 3 minutes. Add in the remaining 2 tablespoons soy sauce and the vinegar, and then add the cabbage, edamame and the reserved chicken, and cook 1 minute longer. Sprinkle in the green scallions.
Bring a large pot of salted water to a boil. Toss in the noodles, swishing them around with a pair of tongs so they don't stick together, and cook until soft, about 30 seconds.
Add the noodles to the pan, stir to combine and cook until the noodles are hot and nicely coated. Add a drop of sesame oil, then transfer to a serving dish and sprinkle with the peanuts and cilantro to serve.