1 3/4 cup plus 2 Tablespoons all-purpose flour (a.k.a. 2 cups minus 2 tablespoons)
2 tablespoons cornstarch
1/4 teaspoon salt
How to make it
In the bowl of an electric mixer fitted with the whisk attachment, cream butter and 1/3 cup sugar until fluffy. Add lime zest, juice, and vanilla; beat until fluffy.
In a medium bowl, whisk together flour, cornstarch, and salt. Add to butter mixture, and beat on low speed until combined.
Between two 8-by-12-inch pieces of parchment paper, roll dough into two 1 1/4-inch-diameter logs. Chill at least 1 hour.
P.S You can freeze these for later use at this point.
Heat oven to 350 degrees. Line two baking sheets with parchment. Place remaining 2/3 cup sugar in a resealable plastic bag. Remove parchment from logs; slice dough into 1/4-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, just three or four minutes. While still warm, place cookies in the sugar-filled bag; toss to coat.