Grease and lightly flour a 9-inch square baking pan.
Whisk flour, ginger, baking soda, cinnamon, salt, and Chinese five-spice powder in a bowl.
Stir sugar, molasses, oil, and egg into flour mixture until just combined.
Pour in boiling water and whisk until the batter is smooth and shiny, about 1 minute.
Pour batter into prepared baking pan. Tap 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.
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 the mixture around with a spatula to ensure even distribution. Let cake cool completely before serving.