In the bowl of an electric mixer, cream butter and 1/3 cup sugar until fluffy. Add lime zest, lime juice and vanilla; beat until fluffy.
In a medium bowl, whisk together flour, cornstarch and salt. Add flour mixture to butter mixture and beat on low speed until combined.
Between two 8-by-12-inch pieces of parchment paper, roll dough into two 1 1/4-inch-diameter logs. Chill until firm, at least one hour.
Preheat oven to 350. Line two baking pans with silpats or parchment paper. Place remaining sugar in a resealable plastic bag. Remove parchment from logs; slice dough into 1/8-inch-thick rounds. Place rounds on pans, spaced 1 inch apart.
Bake cookies until barely golden, 12 to 15 minutes. Transfer cookies to wire rack to cool slightly, 8 to 10 minutes. While still warm, place cookies in the sugar-filled bag and toss to coat.