Whisk chicken broth, cornstarch, and 3 teaspoons soy sauce together in a bowl until completely blended.
Heat 1 tablespoon olive oil in a deep frying pan or wok over high heat. Cook and stir chicken in hot oil until cooked through, 6 to 10 minutes. Remove chicken from pan and drain any accumulated liquids.
Heat remaining 1 tablespoon olive oil in the frying pan or wok over high heat. Stir fry cabbage, snap peas, asparagus, celery, red bell pepper, green onions, and bamboo shoots for 1 minute. Stir in 1 teaspoon soy sauce. Continue cooking until vegetables are tender but still crisp, about 3 minutes.
Stir chicken into cabbage mixture. Pour chicken broth mixture over chicken mixture, reduce heat to medium, and simmer until sauce thickens, about 1 minute. Reduce heat to low; add cashews and cook until heated through, 1 minute. Sprinkle with paprika.