Heat butter in a stock pot over medium heat; add bok choy, mushrooms, green onions, garlic, and ginger. Sprinkle mushroom mixture with lime juice; cook and stir just until lightly browned, 2 to 3 minutes. Increase heat to medium-high, pour in chicken broth, and bring to a boil. Reduce heat to low and simmer until vegetables are soft, about 5 minutes. Serve with cilantro.