In a medium bowl, whisk together flour, cornstarch, and salt. Set aside.
In the bowl of an electric mixer fitted with the whisk attachment, cream butter and ⅓ cup sugar until fluffy. Add lime zest, juice, and vanilla; beat until fluffy. Add dry ingredients, and mix on low speed until blended.
Divide the dough in half. Use two 8-by-12-inch pieces of parchment paper to wrap around and shape each half of the dough into 1 ¼-inch-diameter logs. Chill at least 1 hour
Heat oven to 350 degrees. Line two baking sheets with parchment. Remove parchment from logs; slice dough into ¼-inch-thick rounds. Place rounds on baking sheets, spaced 1 inch apart. Bake cookies until barely golden, about 15 minutes.
Transfer cookies to a wire rack to cool slightly, 8 to 10 minutes. Place remaining ⅔ cup powdered sugar in a resealable plastic bag. While still warm, place cookies in the sugar-filled bag; toss to coat. Store baked cookies in an airtight container for up to 2 weeks.