In a small mixing bowl, whisk together the buttermilk, eggs, and vanilla. Set aside. Combine 1 1/2 cups of the flour, 1/2 cup of the granulated sugar, the baking powder, baking soda, and salt in a separate mixing bowl. Add the buttermilk mixture and melted butter and stir to combine. (The dough will be very sticky.) Allow 5 minutes for the dough to rise. Preheat the oil to 365°F in a heavy deep pot.

Sprinkle 2 tablespoons of the remaining flour on the work surface and 2 tablespoons over the dough itself. Turn the dough onto the work surface and coat the outside of the dough with the flour. Flour your hands and gently press the dough into a 9 X 7 inch rectangle 3/4 inch thick. Cut the doughnuts with a 1 3/4 inch round cookie cutter. Open a hole in the middle of each doughnut with the tip of a knife. Dust your fingertip with flour and increase the opening of the hole. (Doughnuts may also be cut into small rectangles using a knife.)

Check the temperature of the oil and fry the doughnuts in small batches for 45 to 50 seconds. Turn the doughnuts when the dough expands and splits at the top, and cook for an additional 45 to 50 seconds. Transfer the cooked doughnuts to paper towels and cool slightly.

Combine the remaining 1/2 cup sugar and the confectioners' sugar in a bowl. Toss the warm doughnuts in the sugar mixture one at a time. The doughnuts are best eaten warm.