Preheat the oven to 350 degrees F (175 degrees C). Grease and lightly flour a 9-inch square baking pan.
Whisk together flour, ginger, baking soda, cinnamon, salt, and Chinese five-spice powder in a bowl. Stir in sugar, molasses, oil, and egg until just combined. Pour in boiling water; whisk until batter is smooth and shiny, about 1 minute.
Pour batter into the prepared baking pan. Tap the pan gently on the counter to remove any air bubbles.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 35 minutes.
Meanwhile, mix powdered sugar, lemon juice, and lemon zest in a bowl until sugar dissolves.
Pour lemon juice mixture over cake while cake is still hot. Spread mixture around with a spatula to ensure even distribution.