Heat 1/2 teaspoon oil in a skillet over medium heat. Add chicken; cook until golden on the outside and no longer pink in the center, 5 to 7 minutes. Remove from skillet.
Heat remaining 1/2 teaspoon oil in the skillet. Add green bell pepper, onion, celery, and carrot.
Mix boiling water, soy sauce, bouillon cube, salt, and sugar together in a bowl; add to the bell pepper mixture. Bring to a simmer; cook until carrots are tender, about 7 minutes. Add bok choy and bean sprouts. Cook until bok choy is tender, about 5 minutes.
Return chicken to the skillet. Mix 2 teaspoons water and cornstarch together in a bowl; pour into pan and mix. Simmer until broth thickens, 3 to 5 minutes.
