Bring chicken broth to a boil in a stockpot over medium heat. Add green onions; reduce heat and simmer until flavors are released, 1 to 2 minutes. Add bok choy, carrot, celery, soy sauce, ginger, peanut butter, sriracha, garlic, and sugar. Simmer until vegetables are tender, about 10 minutes.
Fill a second large pot with lightly salted water and bring to a rolling boil. Cook ramen noodles in boiling water, stirring occasionally, until tender yet firm to the bite, 2 to 5 minutes. Drain. Portion noodles into serving bowls.
Place chicken into the pot of broth. Simmer until chicken is heated through, about 4 minutes. Pour soup over noodles to serve.