First, make the sauce by mixing together all of the sauce ingredients (peanut butter, lime juice, soy sauce, sesame oil, water, cilantro, brown sugar, fish sauce, lemongrass and garlic) until smooth. This can be done ahead and set aside/refrigerated if kept for longer.
Prepare everything for the stir fry before you start cooking so you can add quickly - slice the onion, slice the eggplant and cut larger slices in half or smaller, as fits, to make small bite-sized pieces. Slice the bok choy and separate the stem part from the leaves. Slice the pepper and cut the broccoli in to small florets as well as chunks of stem. Cut the chicken in to thin slices or strips.
Warm the oil in a stir fry pan or saute pan/skillet. Add the onion and cook for a minute or two then add the eggplant. Cook for a few more minutes until both are softening.
Add the bok choy stems and red pepper and cook for a few more minutes. Add the broccoli and chicken and cook a couple minutes until chicken is cooked through. Add the bok choy leaves, stir so they wilt, then add the sauce. Cook a minute then serve, topped with the peanuts.