All the ingredients should be at room temperature (70 degrees F). Beat the sugar and eggs together until creamy and lemon colored.
Melt the butter and chocolate together in a small saucepan over low heat. Beat into the sugar and egg mixture.
In a separate bowl combine the vanilla and buttermilk. Stir into the sugar mixture.
Combine the flour, baking powder, baking soda, and salt in a large bowl. Stir into the liquid mixture. Chill the dough slightly to make it easier to handle.
Roll out half the dough on a lightly floured board to a 1/2 inch thickness.
Cut into doughnuts. Allow the cut doughnuts to dry for 10 minutes.
Heat 2 inches of oil to 370 degrees F (185 degrees C) in a large skillet. The dough should be at room temperature before frying. Dip a metal pancake turner into the hot oil each time before using it to move or turn a doughnut.
Transfer the doughnuts to the skillet, one every 15 seconds. Fry each doughnut about 90 seconds per side.
Remove the doughnuts and drain on paper towels. Repeat with the remaining dough.
To make the icing: Beat the sugar, egg white, and vanilla together until smooth.
Drizzle onto the doughnuts after they have cooled.
