Preheat the oven to 350°F. Grease a 9×13-inch baking pan and line the bottom with parchment if desired.

Whisk together the flour, baking powder, baking soda, and salt. In a separate large bowl, rub the lemon zest into the sugar until very fragrant.

Whisk the eggs into the lemon sugar, then add the Greek yogurt, oil, lemon juice, and vanilla.

Fold in the dry ingredients just until the batter is smooth and no pockets of flour remain.

Spread the batter evenly in the pan and bake for 27 to 33 minutes, until the center springs back lightly and a toothpick comes out with a few moist crumbs.

Cool the cake completely. Whisk the powdered sugar with 2 tablespoons lemon juice, lemon zest, and salt. Add a little more lemon juice as needed for a thick but pourable glaze, then spread or drizzle over the cooled cake and let it set before slicing.