Cut mushrooms into thin slices.
Line 2 (10-inch) bamboo steamer sections with parchment paper. Place 2 bok choy halves on each paper. Lay a fillet over each pair of bok choy. Sprinkle evenly with ginger; top with mushrooms. Stack steamer sections, and cover with bamboo lid.
Bring to a boil just enough water to submerge bottom rim of steamer in a large, deep-sided skillet. Steam fish 10 minutes or until it flakes easily with a fork.
Meanwhile, heat oils together in a small saucepan over low heat. Transfer cooked fish and vegetables on parchment to serving plates. Drizzle with hot oil mixture, and sprinkle with green onions and cilantro. Serve immediately.