In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium speed until creamy. Add ½ cup of powdered sugar and beat until light and fluffy.
Add 1 tablespoon of lemon zest and 2 tablespoons of fresh lemon juice. Gently blend on low until incorporated, scrapping down sides with a spatula as needed.
In a small mixing bowl combine the flour, cornstarch and salt. Whisk together.
Add the flour in four batches to the creamed butter and lemon mixture. Blend on medium low until each is incorporated.
Place the cookie dough in a small bowl and cover with plastic wrap. Refrigerate for 1 hour.
To bake the cookies, preheat oven to 350°F. Line a baking sheet with parchment paper.
Scoop a level spoonful of dough using a 1-inch cookie scoop. Roll each into a tight ball and place two inches apart on the parchment lined baking sheet.
If you want to make cookie discs, press each dough ball to the desired thickness using the bottom of a flat glass covered with a small piece of wax paper.
Bake for 11-13 minutes or until lightly browned around the edges. Slide the parchment paper and baked cookies onto a wire rack to cool.
Repeat with a new piece of parchment paper placed on a cooled cookie sheet. Cool cookies completely if using a glaze. If covering in confectioners' sugar, allow the hot cookies to cool for a minute then toss in 1 cup of powdered sugar to coat. Cool completely on the wire rack.
To make the glaze, mix together 1 cup of powdered sugar and 2 tablespoons fresh squeezed lemon juice. Add more lemon juice or sugar as needed to reach desired consistency. Spread on each cookie and garnish with lemon zest if desired.
Allow the glaze to fully set then store cookies in a cool place or refrigerator.