In a medium skillet just large enough to hold them in a single layer, cover the chicken breasts with cold water. Add 1 teaspoon salt, set over medium heat and bring to a simmer. Turn the chicken breasts and simmer until just cooked through while remaining juicy, about 5 minutes. Remove from the water and cool to room temperature. Trim off any fat and the central; cartilage and cut the chicken breasts into ½-inch cubes.
Bring a large pot of water to a boil. Add the broccoli and 2 tablespoons salt and cook uncovered, stirring once or twice, until crisp-tender, about 3 minutes (the water need not return to a boil). With a slotted spoon, transfer the broccoli immediately to a bowl of iced water. Add the asparagus to the water and cook, stirring once or twice, until crisp-tender, about 3 minutes. With a slotted spoon, transfer the asparagus to the bowl with the broccoli. When cool, drain the vegetables immediately and pat dry.
Bring a second pot of water to a boil. Add the linguine and 2 tablespoons salt and cook according to the package directions, stirring occasionally , until just tender, about 10 minutes. Drain well, rinse under cold water and drain again. Transfer to a large bowl and toss with the sesame oil. The salad can be prepared to this point several hours in advance. Cover the ingredients and hold at room temperature.
Just before serving, add the dressing, chicken, broccoli, asparagus, sweet pepper and green onion to the bowl with the noodles and toss well. The salad can be prepared to this point up to 1 hour ahead. Cover and hold at room temperature.
Spread the cabbage in a layer on a single large platter or on serving plates. Divide the salad over the cabbage. Sprinkle with the peanuts and cilantro and serve immediately.