LINE a medium bowl with plastic wrap, letting excess extend over sides. Spray pan with cooking spray.
BEAT eggs and sugars in the bowl of a stand mixer fitted with the paddle attachment on medium speed until mixture ribbons and is light in color, 2 to 3 minutes. Add sour cream, 6 tablespoons melted butter, and 2 teaspoons vanilla. Beat on medium speed until well combined, about 2 minutes.
SIFT Swans Down Cake Flour, and lightly spoon into a measuring cup for 4⅓ cups. Add baking powder, 1 teaspoon salt, baking soda, and nutmeg. With mixer on low speed, gradually add flour mixture to egg mixture, beating until well combined. (Dough will be sticky.) Transfer to prepared bowl. Cover and refrigerate until firm enough to roll, at least 1 hour.
ROLL or pat dough to ½-inch thickness on a well-floured surface. Cut dough using a floured 3.5-inch doughnut cutter. Reroll and cut scraps as needed.
HEAT 2 inches of oil in a large heavy-bottomed Dutch oven over medium heat until a deep-fry thermometer registers 350°.
ADD cut doughnuts to oil in batches. Cook until golden brown, about 2 minutes per side. Using a spider strainer, remove doughnuts and let drain on paper towels. Repeat with remaining cut doughnuts. Let cool completely.
STIR together confectioners’ sugar, food coloring, remaining 6 tablespoons melted butter, remaining 1 teaspoon vanilla, and remaining ¼ teaspoon salt in a large bowl. Add enough milk to make a thick icing that ribbons. Spoon sugar mixture into a piping or resealable plastic bag; cut a ½-inch hole in tip or corner.
PIPE icing over top of doughnuts. Garnish with sprinkles, if desired.
TIP: To make doughnuts even easier to handle, refrigerate cut doughnuts on a parchment-lined sheet tray for 20 minutes. Fry and finish as directed.