In a large bowl, beat the butter at medium speed with an electric mixer until creamy. Add powdered sugar, peppermint extract, and salt; beat well. Gradually add flour to butter mixture, beating at low speed just until blended after each addition. Divide dough into thirds; cover and chill for 30 minutes.
Preheat the oven to 325°F. and lightly grease baking sheets.
Working with 1 portion of dough at a time, divide each portion into 12 pieces. Roll each piece into a 2-inch log; curve ends of each log to form a crescent. Place crescents 2 inches apart on baking sheets.
Bake for 18 minutes or until lightly browned. Cool 1 minute on baking sheets. Carefully roll warm cookies in 1 cup powdered sugar, and then cool completely on wire racks. Store cookies in an airtight container.
Notes
Add red or green sprinkles to the rolling sugar to make these look for holiday festive.