Beat butter at medium speed of an electric mixer until creamy. Add 2/3 cup powdered sugar, 1 teaspoon 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 30 minutes.
Working with 1 portion of dough at a time, divide each portion into 12 pieces. Roll each piece into a 2" log; curve ends of each log to form a crescent. Place crescents 2" apart on lightly greased cookie sheets.
Bake at 325° for 18 minutes or until lightly browned. Cool on cookie sheets 1 minute. Carefully roll warm cookies in 1 cup powdered sugar; let cool completely on wire racks.
Combine remaining 1 cup sugar, milk, and 1/4 teaspoon peppermint extract, stirring until smooth. Drizzle icing over cookies and sprinkle with crushed peppermint, pressing gently. Let icing set before serving. Store in an airtight container.