Cream together the butter and 1/3 cup confectioners' sugar using a sturdy fork, a hand mixer, or a stand mixer fitted with the whisk attachment, until pale and fluffy. Add lime zest, lime juice and vanilla, and mix until fluffy.
In a separate bowl, whisk together flour, cornstarch, and salt. Add the dry ingredients to the butter mixture, and mix until just combined.
Divide dough in half. Place each half on an 8x12-inch sheet of parchment paper. Roll in parchment to form a log 1 ¼-inches in diameter. Refrigerate logs until cold and firm, at least 1 hour.
Preheat oven to 350° F. Remove parchment paper from dough, and cut into ¼-inch-thick rounds. Space rounds 1 inch apart on baking sheets lined with silpats or parchment paper. Bake cookies until barely golden, about 13 minutes, rotating sheets halfway through. Transfer cookies to wire racks to cool slightly, 8 to 10 minutes. Place remaining 2/3 cup confectioners' sugar in a shallow bowl or a resealable plastic bag. While still warm, toss cookies in sugar to coat. Cookies can be stored in airtight containers at room temperature up to 2 weeks.