Preheat oven to 350°F (175°C) and line two baking sheets with parchment paper.

In a large bowl, cream together 1 cup softened butter and 1/2 cup powdered sugar until light and fluffy.

Stir in 1/2 teaspoon peppermint extract.

In a separate bowl, whisk together the flour and cornstarch, then gradually blend into the creamed mixture until a soft dough forms.

Roll the dough into 1-inch balls and place them 2 inches apart on the prepared baking sheets.

Bake for 10 to 12 minutes until the bottoms are barely golden; do not brown the tops. Remove from oven and cool completely on a wire rack.

Prepare the frosting by beating together the cream cheese and 2 tablespoons softened butter until smooth and free of lumps.

Gradually add 1 1/2 cups powdered sugar and 1/4 teaspoon peppermint extract, beating until creamy and well combined.

Spread frosting onto the cooled cookies and immediately garnish with crushed peppermint candies.