Add lime zest, lime juice, and vanilla extract, beat until fluffy. Add flour, corn starch and salt to butter mixture, and beat on low speed until combined. Chill until firm, at least 1 hour. Heat oven to 350 . Place remaining sugar in a resealable plastic bag.
Transfer cookies to a plate to cool slightly.
While still warm, place cookies in the sugar-filled bag and toss to coat. Store cookies in an airtight container for up to 2 weeks. Combine the flour, salt and baking soda in a medium bowl. In a large bowl, with an electric mixer, beat the butter, shortening, and applesauce until creamy. Blend in the sugar and vanilla.