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Heat the oven to 375°F with a rack in the middle position. Mist an 8-inch square metal baking pan or glass baking dish with cooking spray, then line it with a 7-by-16-inch sheet of kitchen parchment so the ends overhang. Mist with cooking spray.
In a medium bowl, whisk together the flour, baking powder, baking soda and salt. In a large bowl, combine the sugar and lemon zest. Using your fingers, rub the zest into the sugar until fragrant and the sugar is evenly moistened. Measure out 2 tablespoons sugar; set aside.
Whisk the eggs into the remaining lemon sugar until lightened in color, about 1 minute. Whisk in the oil, lemon juice and vanilla. Gently whisk in one-third of the flour mixture until only a few streaks remain. Gently whisk in half the milk, followed by half of the remaining flour. Whisk in the remaining milk and, finally, the remaining flour mixture.
Pour the batter into the prepared pan and spread in an even layer, then sprinkle evenly with the reserved lemon sugar. Bake until light golden brown and a toothpick inserted at the center comes out clean, 25 to 30 minutes. Cool in the pan on a wire rack for 30 minutes. Using the parchment overhang as handles, lift the cake out of the pan and place it directly on the rack. Cool completely, about 1 hour. Peel off and discard the parchment before slicing.