1. Put the chicken, ginger coins, whole scallions, and 2 teaspoons salt in a medium saucepan. Cover generously with cold water, bring to a boil, then reduce the heat to simmer for 10 minutes. Remove the pot from the heat, cover, and set aside for 5 minutes. Remove the chicken to a plate to cool. When it is cool enough to handle, use your fingers or two forks to shred the meat. If you used thigh meat, you may need to cut the meat into bite-size pieces. Set the chicken aside. (To make ahead, cover and refrigerate the chicken for up to 5 hours.)
2. While the chicken cooks, combine the sesame and peanut oils, vinegar, lime juice, brown sugar, peanut butter, the 1 tablespoon grated ginger, the garlic, shallots, sambal oelek, and ½ teaspoon salt in a blender. Add pepper and additional salt to taste. Set the dressing aside. (To make ahead, cover and refrigerate the dressing for up to 24 hours.)
3. Remove the outer leaves of the cabbage and cut off and discard about 1 inch at the bottom. Remove and discard the core. Rinse the leaves and spin dry, then cut the cabbage in half lengthwise, then crosswise into slices 1/8 to ¼ inch thick.
4. Just before serving, put the cabbage in a large salad bowl along with the peanuts, sesame seeds, cilantro, the sliced scallions, and the shredded chicken. Pour about two thirds of the dressing over the salad and toss gently, adding dressing as needed to lightly coat everything. Mound the salad on dinner plates to serve.