Preheat the oven to 350 degrees F (175 degrees C).

Combine sugar and canola oil in a large bowl; beat with an electric mixer until smooth and creamy. Add molasses and egg and beat until fluffy. Stir in flour, ginger, baking powder, baking soda, and 1 teaspoon lemon zest; mix until well combined. Use a small cookie scoop to measure out teaspoonfuls of dough; place 2 inches apart on an insulated baking sheet and flatten slightly with a spatula.

Bake in the preheated oven until edges just start to brown, 12 to 15 minutes; do not overbake.

Cool on the baking sheet for 1 minute before removing to a wire rack to cool, 8 to 10 minutes more.

Meanwhile, mix together icing sugar, margarine, lemon juice, 1/2 tablespoon lemon zest, and 2 tablespoons milk for the glaze. Mix thoroughly until desired consistency, adding more milk if needed. Drizzle glaze onto cooled cookies.