Cut white stalks from bok choy; slice into 1-inch pieces. Coarsely chop green leaves.
Peel stalks from broccoli; slice 1/4 inch thick. Cut florets into bite-size pieces.
In a large skillet, boil 1/2 cup water. Add bok choy stalks and broccoli; cover. Simmer over medium-low until broccoli is bright green, 5 to 7 minutes. Uncover; cook on high until water evaporates, 2 to 4 minutes.
Add bok choy leaves, oil, and garlic. Cook, tossing often, until garlic is fragrant, 2 minutes.
Press ginger in a sieve over skillet to release juices. Stir in soy sauce.
