In a large bowl of a stand mixer fitted with the paddle attachment, beat the butter and ⅓ cup of the powdered sugar on medium-high speed until light and pale, about 2-3 minutes.
Add the lime zest, lime juice, and vanilla and mix again, scraping down the sides of the bowl.
Add the flour, cornstarch, and salt, mixing until combined. At first it might not seem like the dough will come together, but let it keep mixing and it will form into a nice dough, about the consistency of play-dough.
Divide the dough into 2 equal pieces and roll each half into a 1 ¼-inch diameter log (about 9-inches long each). Wrap tightly in plastic wrap and place in the fridge for 1 hour to chill.
Preheat oven to 350 degrees F. Unwrap the two logs of dough and slice into ¼-inch discs using a sharp knife. Place the disks on parchment-lined baking sheets and bake for 13-14 minutes until barely golden around the edges.
Cool for 8-10 minutes, then transfer half of the cookies to a large ziploc bag along with the reserved ⅔ cup of powdered sugar. Toss gently to coat each cookie with powdered sugar, then remove the cookies to a plate and repeat with the remaining cookies. A few of the cookies will probably break while being tossed with the powdered sugar, but they still taste good!
Store in an airtight container on the counter for up to 2 weeks. Supposedly they taste even better after a week or two since the lime flavor comes through stronger, but I haven't been able to keep any around that long!