Mix flour, salt in small bowl. Beat butter, sugar in large bowl until fluffy, 2 minutes. Beat in egg, mint. On low speed, beat in flour mixture in thirds. Shape dough into disk; divide in half; wrap; refrigerate 1 hour or overnight.
Heat oven to 350°F. Shape dough or cut as below. Place on ungreased baking sheets.
Bake in 350°F oven 11 to 13 minutes, until lightly browned around edges. Transfer cookies to rack; cool.
Prepare Royal Icing.
Paint cookies with thin icing, using thin spatula to spread to edges. Let set. Pipe on thick icing. To Make Candy Canes: Working with half the dough at a time, pinch off pieces, 1 tablespoon each. On floured surface, roll each piece into 9-inch rope. Transfer to ungreased baking sheets, spacing 1 inch apart. Curve into candy canes, Repeat with remaining dough. Chill dough between batches. Bake as in Step 3.
Using half the dough at a time, roll out between waxed paper to 3/16-inch thickness. Remove top sheet; cut shapes; freeze on paper 5 minutes. Transfer cookies to ungreased baking sheets. Bake as in Step 3. Reroll scraps; cut.
For dough, substitute 1/4 cup unsweetened cocoa powder for 1/4 cup flour. Using half the dough at a time, roll out between waxed paper to 3/16-inch thickness; remove top sheet; cut shapes; freeze on paper 5 minutes. Remove cookies to ungreased baking sheets. Press sliced almonds into dough. Bake as in Step 3. Reroll scraps; cut.
For thin icing for base coat, beat together: 1 box (4 cups) confectioners' sugar, sifted; 3 tablespoons powdered egg whites; 8 tablespoons warm water until the consistency of honey, about 6 minutes. Divide among small bowls, tint with food coloring. For thick icing for piping, repeat as above, but using 6 tablespoons warm water; beat until texture resembles sour cream.