Heat the oil in a large skillet over medium-high heat. Season the chicken with 1/4 teaspoon each salt and pepper and cook, tossing occasionally, until browned and cooked through, 4 to 6 minutes. Transfer to a plate.
Add the bok choy and 1/4 cup water to the skillet. Cover and cook until the bok choy is just tender, 3 to 4 minutes.
In a small bowl, combine the soy sauce, barbecue sauce, and scallions. Add to the skillet and bring to a boil. Return the chicken to the skillet and cook, tossing, until heated through, 1 to 2 minutes.