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.
Add the flour, baking powder and salt to the creamed mixture and mix on low speed until completely combined.
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.