Preheat the oven to 350 degrees F (175 degrees C). Grease and lightly flour a 9-inch square baking pan.

Whisk flour, ginger, baking soda, cinnamon, salt, and Chinese five-spice powder together 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, make the lemon glaze: 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.

Let cake cool completely before serving.