Make the cookies: In an electric mixer fitted with the paddle attachment, mix the butter and sugar together until just incorporated. Do not over-mix at this stage, or the cookies may spread while baking.
Add the egg and vanilla extract. Mix again on low speed, stopping to scrape down the sides of the bowl intermittently as needed.
In a medium bowl, whisk together the flour and salt. Add to the butter and egg mixture. Mix on low speed until a dough is formed and there are no longer any streaks of butter in the mixing bowl. The dough will often clump around the paddle attachment while being mixed. This is normal and a good sign that your dough is the right consistency. If your mixture does not come together and is crumbly, add ice cold water 1 tbsp. at a time until the dough clumps. Roll the dough flat between sheets of parchment paper and chill until ready for use, at least 30 minutes.
Preheat the oven to 350°F. Line two baking sheets with parchment paper.
Use a 3.75-inch pig cookie cutter to stamp shapes from the dough, or use the printable PDF as a template (see blog post). Transfer the shapes to the prepared pans and bake for 10 to 12 minutes, or until the cookies are golden brown around the edges. Re-roll dough scraps and repeat process. Transfer cookies to a wire rack to cool completely after baking.
Make the glaze: Whisk the first four ingredients together and 1 tablespoon milk until smooth. The glaze should be very thick but soft enough to settle into a perfectly smooth surface on the cookie. If the glaze is too thick, add additional milk drops at a time until it becomes the correct consistency. Transfer the glaze to a piping bag and pipe it on the cookies, or gently spoon the glaze neatly onto the cookies.
Decorate! While the glaze is still wet, place a mini chocolate chip flat-side-up onto a pig face for the eye. Add holly berries and leaf sprinkles at the neck. Repeat with remaining cookies. Let stand until dry, about 2 hours. Gift peppermint pigs in velvet pouches.