In the bowl of a mixer, combine the butter, lemon zest, and salt. Beat on low just to blend, then let the mixture stand 5 minutes to infuse the butter with the lemon oils.
In a separate bowl, whisk together the flour, cornstarch, baking soda, and baking powder until evenly combined.
Add 1 1/4 cups granulated sugar to the lemon-butter and cream on medium-high speed until pale and fluffy, 2 to 3 minutes, scraping the bowl once.
Beat in the egg, lemon juice, and lemon extract until the mixture looks smooth and slightly thickened, about 30 seconds.
Add the dry ingredients and mix on low just until no dry streaks remain. Avoid overmixing to keep the cookies tender.
Cover and refrigerate the dough for 20 to 30 minutes to firm slightly and control spread.
While the dough chills, heat the oven to 325°F (165°C) and line two baking sheets with parchment paper.
Place the coating sugar in a shallow dish. Scoop level 1 1/2-tablespoon portions of dough (about a #40 scoop), roll lightly into balls, then dip the tops in the sugar. Arrange on the prepared sheets 2 inches apart and gently press each ball to about 1/2-inch thickness.
Bake one sheet at a time for 11 to 13 minutes, rotating the pan halfway through. The edges should look set and the centers pale with tiny surface cracks; they will continue to set as they cool.
Cool on the sheet for 5 minutes, then transfer to a wire rack to finish cooling. Repeat with remaining dough.