Heat oil in a low-sided five-quart saucepan over medium-high heat until a deep-frying thermometer registers 360 degrees. Lightly dust a baking pan with all-purpose flour, and line a second one with paper towels; set both aside.

Meanwhile, place sour cream in a heat-proof bowl on top of a double boiler; set over a pan of simmering water. Heat until warm to the touch. Remove from heat; set aside.

In a large bowl, sift together all-purpose flour, cake flour, granulated sugar, baking powder, baking soda, salt, and nutmeg. Make a large well; place yeast in center. Pour warm sour cream over yeast, and let sit 1 minute.

Place buttermilk, whole egg, egg yolks, and vanilla in a medium bowl; whisk to combine. Pour egg mixture over sour cream. Using a wooden spoon, gradually draw flour mixture into egg mixture, stirring until smooth before drawing in more flour. Continue until all flour mixture has been incorporated; dough will be very sticky.

Sift a heavy coat of flour onto a clean work surface. Turn out dough. Sift another heavy layer of flour over dough. Using your hands, pat dough until it is 1/2 inch thick. Using a 2 3/4-inch doughnut cutter, cut out doughnuts as close together as possible, dipping the cutter in flour before each cut. Transfer doughnuts to floured pan, and let rest 10 minutes, but not more.

Carefully transfer four doughnuts to hot oil. Cook until golden, about 2 minutes. Turn over; continue cooking until evenly browned on both sides, about 2 minutes more. Using a slotted spoon, transfer doughnuts to lined pan. Repeat with remaining doughnuts.

Gather remaining dough scraps into a ball. Let rest 10 minutes; pat into a 1/2-inch-thick rectangle. Cut, let rest 10 minutes, and cook.

When cool enough to handle, sift nonmelting sugar over tops; serve immediately.