Cream butter and 1/3 C. powdered sugar together until fluffy. Add lime zest, lime juice, and vanilla, beat until fluffy.
In a medium sized bowl, whisk together flour, cornstarch, and salt. Add to butter mixture, and beat on low speed until well combined.
Divide dough in half. Roll dough into two logs (1-1/4 inch in diameter), and wrap each log in parchment paper. Chill at least 1 hour. Place remaining 2/3 C. powdered sugar in a resealable plastic bag.
Heat oven to 350º.
Remove parchment paper from logs; slice dough into 1/8 inch thick rounds. Place on baking sheets, lined with parchment paper, space 1-inch apart.
Bake cookies until barely golden, about 15 minutes. Transfer cookies to a wire rack to cool slightly, 8-10 minutes.
While still warm, place cookies in the powdered sugar filled bag; toss gently to coat.
Bake or freeze remaining dough. Store baked cookies in an airtight container for up to 2 weeks.