How to Make It
Adjust oven racks to upper and lower third of the oven. Preheat oven to 350°. Line two baking sheet pans with parchment paper.
In a medium bowl, sift together the flour, baking soda, cream of tartar, baking powder, and salt.
On a large plate, add the 1/3 cup rolling sugar.
To the bowl of a stand mixer fitted with the paddle attachment (or a large bowl if using a hand-held electric mixer), Add the sugar, butter and vegetable shortening. Beat on medium-high until fluffy, about 3 minutes. Scrape down sides of bowl with a spatula. Beat in the egg, buttermilk, lime zest and vanilla. With mixer on low, beat in flour mixture in 2 batches, until well combined.
Roll the dough into small balls about 1 inch in diameter by rolling between your hands. Roll balls through prepared rolling sugar; coating evenly. Space the balls at least 3 inches apart on prepared baking sheets. Do not overcrowd, as the cookies expand while baking. Flatten slightly using the bottom of a glass. Bake, rotating baking sheets midway through baking, until edges are set and just beginning to brown, about 10 minutes. Let cookies cool on baking sheets for 2 minutes, then transfer to a wire rack and let cool completely. Serve.