Sift Dry Ingredients:
In a large bowl, sift or whisk together the cake flour, baking powder, baking soda, and salt.Cream Sugar and Fats:
Using a stand mixer with the paddle attachment, cream the sugar, vanilla, shortening, and melted butter on medium speed for about 3 minutes until light and fluffy. Scrape down the bowl with a rubber spatula.Add Egg Yolks and Buttermilk:
Reduce mixer speed to low and add the egg yolks one at a time, scraping the bowl after each addition. Pour in the buttermilk and mix until fully incorporated.Mix Dough and Chill:
On low speed, add the dry ingredients all at once and mix just until combined. Use a spatula to fold in any remaining flour from the sides. Transfer the sticky dough (similar to a cross between cookie dough and cake batter) to a clean bowl, cover, and refrigerate for 30-60 minutes (at least 30 minutes).Shape and Cut Bars:
Dust a work surface and your hands generously with flour. Place the chilled dough on the floured surface and roll it gently to coat. Pat into a rectangle about 1/2 inch thick and 3 1/2 inches wide, adding more flour as needed. Using a floured bench scraper, cut the rectangle into bars measuring 3 1/2 by 1 1/2 by 1/2 inches. Re-flour the scraper and lightly press it down the center of each bar to create a shallow indent. Brush off excess flour with a pastry brush.Heat Frying Oil:
In a large heavy pot, melt 6 cups of shortening (about 2 inches deep) and heat to 350°F (175°C) over medium-high heat, using a candy thermometer to monitor.Prepare Cooling Rack:
Prepare a cooling rack lined with 4 layers of paper towels.Fry Bars Until Golden:
Once the oil reaches 350°F (175°C), dust a thin metal spatula with flour and carefully lower a bar into the oil, bottom side down. Fry for 1-2 minutes, then flip and fry another 1-2 minutes until golden brown. Transfer to the prepared rack.Cool Fried Bars:
Allow the fried bars to cool completely to room temperature before glazing.