Note: The cookies are softer and more tender when made with unbleached flour that has a protein content of about 10.5 percent. Pillsbury or Gold Medal works best; King Arthur flour has a higher protein content (around 11.7 percent) and will result in slightly drier, cakier cookies. Do not discard the butter wrappers; they have just enough residual butter on them for buttering the bottom of the drinking glass used to flatten the dough balls. To make sure the cookies are flat, choose a glass with a smooth, flat bottom. Rolled into balls, the dough will keep in the freezer for up to 1 week. The baked cookies will keep in an airtight container for up to 5 days.
1. In food processor, process 1/2 cup sugar and 1 teaspoon lime zest until zest is evenly distributed, about 10 seconds; set aside.
2. Adjust oven racks to upper- and lower-middle positions; heat oven to 375 degrees. Line two large baking sheets with parchment paper. Whisk flour, baking powder, and salt in medium bowl; set aside.
3. In standing mixer fitted with paddle attachment or with hand mixer, beat butter, 1 cup granulated sugar, and brown sugar at medium speed until light and fluffy, about 3 minutes, scraping down sides of bowl with rubber spatula as needed. Add egg, remaining 2 teaspoons zest, and vanilla; beat at medium speed until combined, about 30 seconds. Add dry ingredients and beat at low speed until just combined, about 30 seconds, scraping down bowl as needed.
4. Place lime sugar for rolling in shallow bowl. Fill medium bowl halfway with cold tap water. Dip hands in water and shake off excess (this will prevent dough from sticking to your hands and ensure that sugar sticks to dough). Roll heaping tablespoon dough into 1 1/2-inch ball between moistened palms; roll ball in lime sugar, then place on prepared baking sheet. Repeat with remaining dough, moistening hands after forming each ball and spacing balls about 2 inches apart on baking sheet (you should be able to fit 12 cookies on each sheet). Using butter wrapper, butter bottom of drinking glass; dip bottom of glass in remaining lime sugar and flatten dough balls with bottom of glass until dough is about 3/4 inch thick.
5. Bake until cookies are golden brown around edges and just set and very lightly colored in center, 15 to 18 minutes, reversing position of cookie sheets from front to back and top to bottom halfway through baking time. Cool cookies on baking sheet about 3 minutes; using wide metal spatula, transfer cookies to wire rack and cool to room temperature.