Separate the leaves from the stems of the bok choy. Cut the stems into bite-sized chunks and shred the leaves.
Heat the EVOO in large skillet over medium heat. Cook the bok choy stems in the EVOO until slightly tender, about 3 minutes; add the water and leaves and cook until the water evaporates, about 10 minutes more.
Stir in the capers, garlic, and ginger; cook and stir 1 minute more.
Sprinkle the Balsamic Vinegar and lemon juice over the bok choy and remove from heat; serve immediately.