Lime sugar: Remove zest from limes in strips with a vegetable peeler. Cut away white pith from zest. Chop the zest fine, then grind it in a food processor with the sugar until mixture is pale green with flecks of zest. Store leftovers tightly covered in the refrigerator.

Cookies: Beat together butter, shortening, sugar, and 2 tablespoons of the lime sugar with electric mixer until light and fluffy. Beat in egg and vanilla. Sift flour, baking powder and salt together over the egg mixture, then beat on low speed until just combined. Form the dough into a 10-inch log about 2 inches in diameter, then wrap in wax paper. Chill the dough until firm, at least 4 hours. Preheat oven to 375 degrees and set the rack on the middle level. Remove wax paper and cut the log into 1/4-inch-thick rounds. Place cookies at least 1/2 inch apart on ungreased baking sheets. Bake in batches for 10 to 12 minutes, or until pale golden. Transfer immediately with a metal spatula to a rack set over a