Beat eggs in the bowl of a stand mixer until whipped to a creamy color. Mix in sugar gradually until well blended. Mix in buttermilk and vanilla extract. Mix in butter.

Sift flour, baking soda, baking powder, nutmeg, and salt together in a separate bowl. Beat 1/2 of the mixture into the sugar mixture until well blended; beat in remaining 1/2. Let batter rest for 20 minutes.

Heat oil in a deep-fryer or large saucepan to 375 degrees F (190 degrees C).

Turn dough out onto a floured surface and knead as little as possible until it comes together. Roll dough out using a rolling pin until it is 3/8-inch thick; cut into donut shapes.

Lower donuts carefully into the hot oil and cook until light brown, 2 to 3 minutes, turning once. Remove from oil and let cool on a wire rack.