Preheat the oven to 350°F (175°C) and prepare an 8-inch round cake pan.
Rub the lemon zest into the sugar until fragrant.
Whisk together the flour, baking powder, baking soda, and salt.
In another bowl, whisk together the oil, egg, yogurt, milk, lemon juice, and vanilla.
Fold the wet ingredients into the dry ingredients until just combined.
Spread the batter evenly into the prepared pan.
Bake for 25–30 minutes, or until a toothpick comes out clean.
Cool in the pan for 20 minutes, then transfer to a wire rack.
Whisk together the glaze ingredients until smooth.
Pour the glaze over the cooled cake and finish with extra lemon zest if desired.