Place the soft butter in a mixing bowl. Stir with a wooden spoon or spatula until nice and smooth. Add sugar and vanilla. Mix together by hand until fluffy and well blended.
Add flour and cornstarch. Stir until flour is incorporated. Turn out onto a lightly floured surface and press dough into a ball.
On a lightly floured work surface, roll out the dough to a 3/8-inch thickness. Keep the surface and rolling pin lightly dusted with flour. Cut desired shapes and place onto the baking sheets. Re-roll all the scrapes until the dough has been used up.
Place the cutouts in the refrigerator for at least 1 hour.
When ready to bake, turn your Pit Boss grill to smoke mode, let the fire catch and then set to 350 degrees F (177 degrees C) to preheat. Bake cookies for 12-14 minutes or until just beginning to turn golden at the edges. Cool completely before icing.
For the mint glaze, combine powdered sugar, milk and peppermint extract in a bowl. Mix until smooth. Glaze should be thick but pourable.
To glaze the cookies, dip the top of the cookie into the glaze. Allow extra glaze to drip back into the bowl. Quickly flip the cookie right side up. Allow the glaze to dry for 15-30 minutes before serving. Enjoy!