In a large mixing bowl with an electric mixer, combine 1/2 cup of the powdered sugar and the butter and mix until light and fluffy. Add the peppermint extract and mix to combine. Next, slowly add the flour and salt, mixing until just combined.
In a separate mixing bowl, combine remaining 1/2 cup powdered sugar and the peppermint candies.
Roll the dough into twenty-four 1 1/2-inch balls. Insert 1 chocolate morsel into each ball and re-roll to ensure that the cookie dough encases the chocolate. Next, transfer to the bowl with the powdered sugar and crushed candies rolling each to coat.
Place the balls on an ungreased baking sheet, spacing them about 1-inch apart. Bake until they start to turn golden brown on the bottom, about 20 minutes. Remove and transfer to wire cooling rack. Cool until they are just cool enough to handle, 5 minutes. Return the cookies to the bowl with the powdered sugar and crushed candies and toss gently until each cookie is coated.