Preheat oven to 375 degrees. Line a baking sheet with parchment paper.

In a large bowl, combine the butter and sugar and beat on medium speed using a stand or hand mixer until light and fluffy, about 3-4 minutes. Add the egg, 1 tablespoon of lemon zest, 1 teaspoon of lemon juice and the vanilla, and beat until creamy, another minute or two.

Divide the dough into 12 even-sized portions. I just use a spoon to roughly divide the dough into 4 sections in the bowl, then try to scoop three equal-sized scoops from each section so that I end up with 12 cookies that are the same size. Roll each section of dough into a ball and set on the parchment-lined cookie sheet, spaced evenly apart.

Using the bottom of a glass cup, first get it a little wet by running it under water, and then dip it in granulated sugar. Use the sugar-coated bottom of the glass to press down gently on each ball of dough, flattening it slightly until they are about 3 inches in diameter. You don't want the cookies super thin but they don't spread much when baking so pressing them down helps them bake evenly.

Bake for 12-14 minutes or just until the cookies look set, then remove from the oven and cool on the cookie sheet for 2 minutes before transferring them to a wire rack to cool completely.

While the cookies are cooling, prepare the lemon glaze by combining the powdered sugar, lemon zest, and enough lemon juice to get a barely runny consistency. You don't want it too runny or you will just have a super thin layer of glaze on top of the cookies, but you want it a little more runny than a frosting that you would spread. I add lemon juice just until the glaze starts to just barely drizzle off my mixing spoon when I hold it up over the bowl.

Transfer the glaze to a Ziploc bag (I find that the freezer ones work better for this than the thin, sandwich variety), seal the bag, and snip off a small corner to create a piping bag. Pipe or drizzle the glaze over the cooled cookies on the wire rack in a zigzag motion, allowing some of the glaze to drip over the edges. Let the cookies stand just until the glaze is set before serving.