1. In the bowl of an electric mixer fitted with the whisk attachment, cream butter and 1/3 cup sugar until fluffy. Add lime zest, juice, and vanilla; beat until fluffy.
2. In a medium bowl, whisk together flour, cornstarch, and salt. Add to butter mixture, and beat on low speed until combined.
3. Between two 8-by-12-inch pieces of parchment paper, roll dough into two 1 1/4-inch-diameter logs. Chill at least 1 hour.
4. Heat oven to 350 degrees. Line two baking sheets with parchment. Remove parchment from logs; slice dough into 1/8-inch-thick rounds. Place rounds on baking sheets, spaced 1 inch apart.
5. Bake cookies until barely golden on the edges, about 8-10 minutes. Transfer cookies to a wire rack to cool slightly, 8 to 10 minutes. Place remaining 2/3 cup sugar in a resealable plastic bag. While still warm, place cookies in the sugar-filled bag; toss to coat. Bake or freeze remaining dough. Store baked cookies in an airtight container for up to 2 weeks.

Makes 6 Dozen Cookies.