Preheat oven to 400 degrees F (200 degrees C). Cream together butter, shortening, sugar, egg and the vanilla.There is something that hand mixing does this to this cookie. I tried this recipe twice - once when I mixed it using a hand mixer and once while with hand. I dunno what exactly - but I think hand mixing made the texture all the more better for this cookie. May be they looked puffier and prettier when done by hand. You be the judge.:)
In a separate bowl, blend in the flour, cream of tartar, soda and salt.
Now add the dry into the wet ingredients and blend well. Make sure you don't overmix them. The result will be a slightly sticky dough.
Mix the cinnamon, cardamom (if using) along with sugar and spread it on a plate. Shape dough by rounded spoonfuls into balls. Since the dough is little sticky, I actually dusted my hands with the sugar+spice mixture to help me shape the rounds. You anyways have to coat them with this mixture and I thought why not use it to make rounds too. It worked pretty well for me. Once shaped, roll the balls of dough in the sugar+spice mixture until well coated on all sides. Make sure to place these balls at least 2 inches apart on an ungreased baking sheet (or use parchment paper). They spread while baking. Depending on the size, you get about 22-24 cookies with this recipe.
The original recipe states to bake these for 8-10 minutes. I personally found 5 minutes worked out perfect for me in my Oven. So keep a look out from 5 minutes onward. The time to remove would be when you cookies have spread out, have the characteristic texture of the snickerdoodle, i.e cracked surface and have very slightly browned on top but might look still soft in the edges. Its fine since they will hardene up while cooling. Otheriwse they might turn out too hard.
Remove them from the oven, let them sit for few seconds and then take out the cookies from the pan.