Line two baking sheets with parchment paper and set aside. In a medium bowl, stir the softened butter until completely smooth and creamy. Add the 1/2 cup powdered sugar and vanilla extract, mixing for about 1 minute until well combined. In a separate small bowl, whisk together the all-purpose flour and cornstarch. Add the dry mixture to the butter mixture and stir until the dough comes together in a shaggy consistency. Knead gently with your hands until the dough forms a cohesive disk, then wrap it in plastic wrap and refrigerate for at least 30 minutes—this helps the dough become easier to roll and prevents excessive spreading during baking.

Remove the dough from the refrigerator and place it between two sheets of parchment paper. Roll the dough to 3/8-inch thickness using a rolling pin—the parchment makes cleanup easier and prevents sticking. Remove the top sheet of parchment and cut out your desired Christmas shapes using cookie cutters. Carefully transfer the cut-outs to the prepared baking sheets, spacing them about 1 inch apart. I like to chill these for at least 2 hours before baking (or up to 24 hours) because this prevents the cookies from spreading too much in the oven and helps them maintain their shape.

Preheat your oven to 350°F. Bake the chilled cookies for 12-16 minutes, until the edges are light golden brown but the centers still look slightly underbaked—they will continue cooking on the hot pan after removal. Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely before decorating. This cooling time is crucial because it allows the glaze and frosting to adhere properly.

In a small bowl, whisk together the 2 cups sifted powdered sugar, 4 tablespoons half and half, and 1/4 teaspoon peppermint extract until smooth and pourable—add a splash more half and half if needed to reach a glaze consistency. Dip the top of each cooled cookie into the glaze, allowing excess to drip back into the bowl. Arrange the glazed cookies on a clean parchment paper or wire rack and let the glaze set for 15-30 minutes before adding the buttercream decoration.

In a small bowl, stir the 1 tablespoon softened butter until smooth and creamy. Add the 1 cup powdered sugar, 1.5 tablespoons half and half, and 1/2 teaspoon peppermint extract, stirring until the frosting reaches a piping consistency—it should be thick enough to hold its shape when piped but soft enough to come through a pastry bag. Add 3 drops of green food coloring and stir until the color is evenly distributed. Transfer the buttercream to a pastry bag fitted with a small round or leaf tip, then pipe decorative Christmas trees, garlands, or other festive designs onto the glazed cookies. Add sprinkles immediately if desired while the frosting is still wet, then let everything set for at least 30 minutes before serving or storing.