Preheat oven to 350 degrees Fahrenheit and line two baking sheets with parchment paper or a silicone baking mat.
Beat the butter on medium speed using a hand or stand mixer until it is light and fluffy.
Turn the mixture to low speed and add in the powdered sugar and lemon zest. Mix until well incorporated, scraping down the sides of the bowl as needed.
With the mixer on low speed, gradually add in the flour.
Next add in milk and lemon extract.
Keep mixing until a shaggy dough forms - about 1 minute. Form the dough into a disk and wrap it in plastic wrap. Refrigerate for 15 minutes (this will help make and dough easier to roll and cut).
On a lightly floured surface roll out the dough until it is 1/4" thick.
Using a cookie cutter of your choice, cut out the cookies and place them on the baking sheets.
Place the sheets in the freezer for 10 minutes to chill the dough again (this prevents spreading). Bake until the edges of the cookies are golden brown - about 12 to 14 minutes. Transfer to a wire rack to cool and then store in a plastic container at room temperature for up to 3 days.
Notes
Storing: Store cooked and cooled lemon shortbread cookies in a plastic airtight container at room temperature for up to 3 to 4 daysFreezing: To freeze cooked and cooled shortbread cookies, place shortbread cookies in an airtight container and place them in the freezer for up to 3 months.Chocolate Topping: You can drizzle the cookies with melted white chocolate, if desired.