Line a 9×13-inch metal pan with parchment, leaving an overhang for lifting. Preheat the oven to 350°F (175°C).
In a small bowl, rub the lemon zest into the granulated sugar with your fingertips until the sugar feels slightly damp and intensely fragrant. This blooms the citrus oils.
In a large bowl (or stand mixer fitted with the paddle), cream the softened butter with the lemon sugar on medium speed until very pale and fluffy, 2 to 3 minutes. Scrape down the bowl as needed.
Beat in the eggs one at a time until fully incorporated, then mix in the lemon extract just until combined.
Whisk together the flour, cornstarch, baking soda, and salt. Add the dry ingredients to the butter mixture in two additions on low speed, mixing only until no dry spots remain. Finish by folding once or twice with a spatula to ensure an even, soft dough.
Scrape the dough into the prepared pan and press it into an even layer with an offset spatula. Chill the pan in the freezer for 10 minutes to help the bars bake evenly and retain a soft center.
Bake on the center rack for 21 to 24 minutes, rotating once halfway through. The edges should be lightly golden and the center should spring back when gently touched. Do not overbake for the softest texture.
Cool the pan on a rack until the bars are completely cool, about 1 hour. Lift out using the parchment overhang.
Make the frosting: In a stand mixer with the paddle or a bowl with a hand mixer, beat the butter on medium speed until creamy and slightly lightened, about 2 minutes. Add half the powdered sugar and mix on low until combined. Blend in the lemon juice, lemon zest, vanilla, and a pinch of salt. Add the remaining powdered sugar, then beat on medium-high for 2 to 3 minutes, adding 2 to 3 tablespoons cream a little at a time until fluffy and spreadable.
Spread the frosting over the cooled slab. For the cleanest slices, chill 15 minutes to set, then cut into 24 bars. Garnish with a touch of extra zest if you like and serve.