Combine first 6 ingredients in a bowl, stirring with a whisk. Set aside.
Heat 2 teaspoons oil in a large nonstick skillet or wok over medium-high heat. Add onions, ginger, and garlic; stir-fry 2 minutes. Add chicken; stir-fry 5 minutes. Remove chicken mixture from pan. Add 1 teaspoon oil to pan. Add bok choy, bell pepper, and water chestnuts; stir-fry 3 minutes. Add chicken mixture and peanut butter mixture to pan; stir-fry 2 minutes. Serve over rice; sprinkle with peanuts.