In a stand mixer, fitted with a paddle attachment, beat the sugar, shortening, and egg yolks until smooth and light in color, about 2-3 minutes. Add the salt, sour cream and vanilla and beat until smooth.
Mix 3½ cups of flour, cornmeal, baking powder, nutmeg, cardamom, and xanthan gum in a medium bowl until combined.
Add the flour mixture to the sour cream/egg mixture and beat until it comes together. It should be slightly tacky to the touch. If it's very sticky still, add more flour by the tablespoon until it is the consistency of cookie dough.
Lightly flour a parchment lined baking sheet. Scoop the dough, about ⅓ cup at a time and roll it on the floured surface into a snake, about 6-inches long. Form the snake into a ring, pressing the ends together to seal. Repeat with the remaining dough.
Fill a large sauce pan with 3 inches of canola oil (or other frying oil). Heat oil to 350 degrees (test with a candy thermometer). Gently drop a few rings into the oil. They will sink to the bottom, then bounce to the top of the pan. Let them cook for about 20 seconds, or until lightly golden brown. Flip over and let cook for 90 seconds. Flip again, and cook for another 30 seconds. The doughnuts may crack open a little.
Remove the doughnuts from the oil with a slotted spoon and transfer to a cooling rack. Cool for at least 10 minutes.
Mix the sugar and cinnamon in a small bowl. Turn each doughnut over in the sugar until well coated. Serve immediately.