In a stand mixer or a large bowl with hand beaters, beat the cold butter, oil, and sugars on low for 30 seconds. Scrape the bowl, then increase to medium speed for another 30 seconds, then high speed for 30 seconds, or until very light and fluffy.
¾ cup (1.5 sticks) butter, ¼ cup canola oil, 1 cup granulated sugar, ½ cup powdered sugar
Beat in the eggs, one at a time. Add the vanilla. Scrape well.
2 eggs, 1 teaspoon vanilla
In a medium bowl, combine the cornstarch, baking soda, salt, cake flour, and all-purpose flour. Add the flour mixture to the butter mixture and combine on low until just combined. The dough should be very moist. Scrape the bowl well.
1 teaspoon cornstarch, 1 teaspoon baking soda, ½ teaspoon kosher salt, 1 ½ cups cake flour, 2 cups all-purpose flour
Using a 3.25 oz cookie scoop (or ⅓ cup plus 1 tbsp), scoop cookies onto baking sheets lined with parchment paper. Put no more than 6 cookies on each sheet.
Use a flat measuring cup to gently flatten the cookies to about ¾ inch thick. Then chill in the fridge for 30 minutes.
Once chilled, preheat the oven to 400°F. Bake one cookie sheet at a time for 9-10 minutes, until the cookies are slightly golden on the bottom. Cool completely on the cookie sheet.