In a large mixing bowl and using a hand mixer, beat butter and sugar till it is light and fluffy, about 4-5 minutes. Using a spatula, scrape the sides of the bowl, bringing the butter, sugar mixture back to the middle. Add the egg, egg white, and vanilla, then continue beating until combined.
In a medium-size bowl, mix the flour, baking powder, and salt to combine. Sift the flour mixture into the wet ingredients and stir until just combined.
Cover with plastic wrap and refrigerate for at least 20 minutes.
In a small bowl combine the sugar and cinnamon. Using a spoon or cookie scoop, roll the dough into small balls until they are round and smooth. Drop and roll the balls into the cinnamon-sugar mixture and coat well till they are completely covered.
Place them on a baking sheet, lined with parchment paper. Bake for about 9 minutes,* just till the edges are set and golden.
Remove the snickerdoodles from the oven and allow them to sit on the pan for 5-10 minutes before moving to a cooling rack.
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Notes
*the cookies will seem underbaked when cooked at first. This is normal and will help keep the normal snickerdoodle texture you're used to when using cream of tartar