Cut the chicken into 1-inch cubes. Split the dried chili in half lengthwise. Heat the oil in a wok or large skillet, and add the chili. (You may remove it when it turns black or leave it in.)
Add the chicken cubes and peanuts and stir-fry them for 1 minute. Remove the chicken, peanuts, and chili from the pan.
Put all the sauce ingredients, except the sesame oil, into the pan. Bring the sauce to a boil, and then turn the heat down. Return the chicken and peanuts to the pan and cook for about 2 minutes in the sauce. Add the sesame oil and serve immediately.