Warm, homemade doughnuts make a decadent birthday treat.
Clip a deep-fry thermometer on the side of a low-sided 6-quart saucepan, and add the oil. Place pan over medium-high heat; heat oil until the thermometer reads 375 degrees. Lightly dust a baking sheet with flour, and line another one with paper towels; set both baking sheets aside.
Meanwhile, place sour cream in a heat-proof bowl set over a pan of barely simmering water; heat until sour cream is warm to the touch. Remove from heat.
In a large bowl, sift together 2 cups all purpose flour, the cake flour, sugar, baking powder, baking soda, salt, and nutmeg. Make a large well in the center; sprinkle yeast in well. Pour warm sour cream over yeast; let sit 1 minute to activate.
Place buttermilk, whole egg, egg yolks, and vanilla in a medium bowl; whisk to combine. Pour egg mixture over sour cream in well. Using your hands or a wooden spoon, gradually draw flour mixture into the egg mixture, stirring until smooth before drawing in more flour mixture. Continue until all flour mixture has been incorporated and dough has come together. Dough will be very sticky.
Sift a heavy coat of flour onto a clean work surface. Turn out dough onto flour; sift another heavy coat of flour over dough. Using your hands, pat dough until it is 1/2 inch thick. Using a 2 3/4-inch doughnut cutter dipped in flour, cut out as many doughnuts as possible, dipping cutter in flour each time. Transfer to floured baking sheet; let rest 10 minutes but not longer.
Working in batches of four, carefully place doughnuts in the hot oil. Cook until surface is golden, about 1 minute. Turn over; continue cooking until surface is evenly browned, about 1 minute. Using a slotted spoon, transfer doughnuts to paper-towel lined baking sheet, and drain. Adjust heat as needed to maintain a steady 375 degree temperature. Repeat with remaining doughnuts.
Gather dough scraps together, and form into a ball; let rest 10 minutes. Using your hands, pat dough into a 1/2-inch-thick round. Cut into doughnuts and cook following the processes in steps 5 and 6.
While doughnuts are cooling, make the glaze: Place chocolate chips in a medium heat-proof bowl. Bring cream to a boil in a small saucepan, and pour over chocolate. Whisk until smooth, thickened, and combined. Dip top half of each doughnut into glaze; let excess fall back into pan. Quickly invert so that glaze is on top, and place on a wire rack. Immediately decorate with sprinkles, if desired. Let glaze set, and serve.
If time is an issue, you can glaze and decorate store-bought doughnuts instead.