Preheat oven to 350 degrees F.
Using a stand mixer with the paddle attachment, cream together the butter and sugar until light and fluffy—about 3-5 minutes. Slowly add the egg, egg yolk, vanilla, and lemon zest and mix on low, scraping the bowl as needed to make sure it is all mixed.
In a separate large bowl, combine the flour, cornstarch, baking soda, and salt. Mix until combined, then add the flour mixture to the butter mixture and mix on low until the dough is well-combined. Do not overmix.
Using spoons or a small ice cream scoop, scoop 1" balls, place them onto a parchment-lined baking sheet about 3" apart and then flatten them slightly with the palm of your hand.
Bake in a preheated oven for 9-11 minutes, or until slightly golden on the edges. Remove from the oven and let cool on the baking sheet for a couple minutes, then transfer them to a wire rack to cool completely.
Meanwhile, in a small mixing bowl combine the powdered sugar, lemon juice and lemon zest. Well until completely smooth, adding a little extra lemon juice or water to thin down as necessary.
Use a spoon to drizzle the cooled cookies with the glaze, then let the glaze set at room temperature before storing. Enjoy!