Either use pre-ground cardamom seeds or—better still—grind the cardamom seeds fresh. If using green cardamom pods, discard the papery green sheaths before grinding the dark seeds.
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, ground cardamom, baking soda, cream of tartar, baking powder, and salt.
On a large plate, combine the 1/3 cup rolling sugar and cinnamon on a plate.
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 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.
The dough can be refrigerated for up to three days or frozen for up to two months.