Heat chicken broth in a stock pot over medium heat until boiling. Add green onions; reduce heat and simmer until flavors are released, 1 to 2 minutes. Add bok choy, soy sauce, carrot, celery, ginger, peanut butter, sriracha, garlic, and sugar. Simmer until vegetables are tender, about 10 minutes.

Fill a large pot with lightly salted water and bring to a rolling boil; stir in ramen and return to a boil. Cook, uncovered, stirring occasionally, until the ramen is 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 the noodles.