Put the peanut oil in a heavy skillet large enough to hold all the chicken in 1 layer without crowding. Put the skillet over medium-low heat. Spread the sesame seeds on a pie plate or waxed paper. Season the breasts with salt and pepper and press into the sesame seeds to coat both sides evenly. Put in the skillet, raise the heat to medium-high, and cook for 5 minutes on each side, gently turning the breasts with a spatula, until the meat is firm and no pink shows in the middle. Reduce the heat if needed to prevent burning.
Meanwhile, trim the bottom 1/2-inch from the cabbage and remove any damaged or withered parts from the tops of the leaves. Cut crosswise into strips no more than 1/2-inch wide. Put into a salad spinner, fill with water, drain and spin the greens dry. Remove any excess moisture with paper towels.
Peel and halve the ginger. Drop down the chute of a food processor with the motor running and puree. Stop the motor and scrape down the sides of the bowl with a rubber spatula. Add the fish sauce, vinegar, and sesame oil. Juice the lime and add the juice to the ginger mixture. Process until the dressing is combined. Add salt to taste.
Open the can of water chestnuts and drain. Trim the scallions and cut the white and green parts crosswise into thin slices. Cut the top from the bell pepper. Stand it upright and cut down inside the four walls, separating them from the center core and seeds. Then cut the walls into thin strips. Coarsely chop the leaves from the cilantro sprigs.
Put the cabbage in a large mixing bowl. Add the water chestnuts, scallions, bell pepper, cilantro, and dressing. Toss well. Cut the cooked chicken into strips no more than 1/2-inch wide. Add to the salad and toss.