3/4 cup (1 1/2 sticks) unsalted butter at room temperature
1 cup powdered sugar
Finely grated zest of 2 limes
2 tablespoons fresh lime juice
1 tablespoon pure vanilla extract
1 3/4 cups plus 2 tablespoons all-purpose flour
2 tablespoons cornstarch
1/4 teaspoon salt
Put butter and 1/3 cup confectioners’ sugar in the bowl of an electric mixer fitted with the whisk attachment, and mix on medium speed until pale and fluffy. Add lime zest and juice and vanilla, and mix until fluffy.
Whisk together flour, cornstarch, and salt in a bowl. Add to butter mixture, and mix on low speed until just combined.
Divide dough in half. Place each half on an 8-by-12-inch sheet of parchment paper. Roll in parchment to form a log 1 1/4 inches in diameter. Refrigerate logs until cold and firm, at least 1 hour.
Preheat oven to 350 degrees. Remove parchment from logs; cut into 1/4-inch-thick rounds. Space rounds 1 inch apart on baking sheets lined with 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. While still warm (but not piping hot), toss cookies with remaining 2/3 cup sugar in a resealable plastic bag. Cookies can be stored in airtight containers at room temperature up to 2 weeks.