Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper.
In a large bowl, beat the softened butter on medium speed with a hand or stand mixer until light and fluffy, about 1–2 minutes.
Reduce the mixer to low speed and add the powdered sugar and lemon zest. Mix until well incorporated, stopping to scrape down the sides of the bowl as needed.
With the mixer still on low, gradually add the all-purpose flour and mix until combined.
Add the milk and lemon extract and mix on low until a shaggy dough forms, about 1 minute.
Turn the dough out onto a lightly floured surface, gently bring it together if needed, form it into a disk, wrap in plastic wrap, and refrigerate for 15 minutes to firm.
On a lightly floured surface, roll the chilled dough to 1/4-inch thickness. If the dough softens while working, chill it for 5–10 minutes before continuing.
Use a cookie cutter to cut out shapes and transfer the cookies to the prepared baking sheets, spacing them about 1 inch apart. (Using a round cutter as in the original produced about 10 cookies.)
Place the baking sheets in the freezer for 10 minutes to chill the dough and help prevent spreading.
Bake the chilled cookies at 350°F for 12–14 minutes, or until the edges are golden brown.
Let the cookies cool on the baking sheets for a few minutes, then transfer them to a wire rack to cool completely.
Store cooled cookies in a plastic container at room temperature for up to 3 days.