1. Whisk together flour, cornmeal, baking soda, cream of tartar, and salt in a large bowl. Set aside.
2. Beat butter, shortening, and 1 cup sugar in a large mixer bowl on medium-high speed until light and creamy, 2 to 3 minutes. Beat in corn syrup, then egg until combined. Beat in vanilla and 3 tablespoons lime peel, scraping side of bowl until blended. At low speed, beat in flour mixture just until combined. Cover and refrigerate dough 1 hour.
3. Heat oven to 350 degree F. Mix together 2 tablespoons sugar and remaining 1/2 teaspoon lime peel in a small bowl until well combined. Set aside.
4. Spread remaining 1/3 cup sugar on a plate. Line 2 cookie sheets with foil. Fill a 1/4-cup dry measuring cup with dough. Shape dough into a ball; then roll in sugar. Place on prepared cookie sheet. With bottom of a glass, gently press top of ball into a 2 1/2-inch circle. Repeat with remaining dough and sugar, arranging 4 cookies on each cookie sheet. Bake cookies until edges begin to color, 12 to 15 minutes. Cool cookies on pan 5 minutes. Sprinkle warm cookies with lime-sugar mixture. Transfer to wire racks and cool completely.