In a stand mixer or a large mixing bowl, combine the cold butter, canola oil, and both sugars. Begin mixing on low speed for about 30 seconds, scrape down the sides, then increase the speed to medium for another 30 seconds. Finally, mix on high speed for 30 seconds until the mixture is fluffy and well-aerated.

Add the eggs one at a time, mixing well after each addition. Pour in the vanilla extract and mix until fully incorporated. Scrape the bowl to ensure everything is well blended.

In a separate medium-sized bowl, whisk together the cake flour, all-purpose flour, cornstarch, baking soda, and kosher salt. Gradually add the dry ingredients to the wet ingredients. Mix on low speed just until combined, creating a soft and moist dough. Scrape down the bowl to make sure no pockets of flour remain.

Using a 3.25 oz cookie scoop or measuring about ⅓ cup plus 1 tablespoon per cookie, portion the dough onto parchment-lined baking sheets. Space the cookies generously, with no more than six per sheet.

Gently flatten each dough ball to about ¾ inch thickness using the base of a flat measuring cup. Place the baking sheets in the refrigerator to chill for 30 minutes.

While the cookies chill, preheat the oven to 400°F. Bake each sheet individually for 9–10 minutes, or until the bottoms are lightly golden. Leave the cookies to cool completely on the baking sheets before handling.

As the cookies cool, prepare the buttercream. In a large bowl, beat the softened butter and cream cheese on medium speed until the mixture becomes light and fluffy.

Reduce the speed to low and add half of the powdered sugar. Mix until it begins to come together, then add the remaining powdered sugar. Increase the speed to medium and beat for about a minute.

Incorporate the vanilla extract and ½ tablespoon of heavy cream. Beat the mixture for another 2 minutes until it reaches a smooth and airy consistency. If needed, add additional cream in small increments until the frosting is soft enough to pipe while still holding its shape.

Fill a large piping bag fitted with a 2A tip, or use a resealable plastic bag with a small corner snipped off. Pipe the frosting in a spiral pattern onto each cooled cookie. Top with festive Christmas sprinkles to finish.