Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a mixing bowl, whisk together the all-purpose flour, baking powder, and salt. Set aside.
In another mixing bowl, beat the softened butter and granulated sugar with an electric mixer until creamy (about 2-3 minutes).
Add the egg, vanilla extract, and peppermint extract to the butter mixture, mixing until well combined.
Gradually add the flour mixture to the wet ingredients, stirring until a dough forms.
Lightly flour a surface and roll out the dough to about ¼ inch thickness.
Use a tree-shaped cookie cutter to cut out tree shapes and place them on the prepared baking sheet.
Bake in the preheated oven for 8-10 minutes or until the edges are lightly golden.
Allow the cookies to cool on a rack for about 10 minutes before transferring them to the cooling rack.
For the frosting, mix the powdered sugar, milk, and peppermint extract in a bowl until smooth.
Once the cookies have completely cooled, frost each tree and sprinkle with crushed peppermint candies for decoration.
Let the frosting set before serving.