Preheat oven to 350 degrees F. Line two cookies sheets with parchment paper.
In a large bowl of a stand mixer fitted with the paddle attachment, add the butter, oil, granulated sugar and powdered sugar. Sprinkle the baking soda, cream of tartar and salt across the top of the sugars evenly. Mix on medium speed until well-combined and creamy about 1-2 minutes. Scrape down the sides of the bowl as needed.
Next, add the sour cream, eggs and vanilla. Continue mixing over medium speed for an additional 1-2 minutes until combined.
Slowly add the all purpose flour and beat until no pockets of flour remain. Don't over mix.
Scoop the dough into about 3-tablespoon sized portions and roll into balls. Place several inches apart on the prepared baking sheets. Add ½ cup of sugar into a bowl and lightly spray the bottom of a glass with cooking spray. Dip the bottom of the glass into the sugar and press each cookie into an even thickness, about ½ inch thick.
Bake the cookies until just set, about 9 minutes. Don't allow the cookies to brown along the edges or they will be dry. They may appear under baked- but they will firm up and be nice and soft once cooled. Repeat rolling, pressing and baking until there is no more cookie dough.
Set the cookies aside on cooling racks to cool completely while you start on the peppermint twist frosting.
In a large bowl, beat together the softened butter and cream cheese until completely smooth. Add the vanilla, peppermint extract (or peppermint essential oil), and salt. Beat well to combine. Next add the powdered sugar and then beat until the frosting is smooth, about 3 minutes.
When the cookies are completely cool, take about a tablespoon dollop of frosting on each cookie and spread with a spoon, spinning the spoon as you go to create a swirl if desired. Top with crushed candy canes. ENJOY!