Heat oven to 350℉.
Line 3 sheet pans with parchment paper. Set aside.
With an electric mixer, beat butter and sugar just until combined, about 1-2 minutes.
Beat in egg and egg yolk.
Stir in flour, salt, and baking powder just until mixed.
Mix in milk and peppermint extract.
Divide dough in half.
Wrap half in plastic wrap and set aside.
On a non-stick surface (marble, parchment, silicone), roll other half of dough into a rectangle that is 3/16 inch thick.
Use a cookie cutter to cut shapes as closely together as possible.
Remove excess dough.
Place cut dough, still on parchment, on a baking sheet.
Freeze for 10 minutes.
Remove cutouts from parchment and place on prepared pans.
Bake for 15 minutes, until edges just start to brown.
Allow to cool on pans.
Repeat with other half of dough.
Gather and re-roll scraps. Repeating the steps above until no more dough remains.