In a large mixing bowl, combine the flour, salt, and sugar. Set aside.
In a small bowl, dissolve the yeast in warm water and let the mixture sit for about 5 minutes, or until foamy.
Make a well in the center of the dry ingredients and pour the buttermilk, melted butter, egg and the yeast mixture. Use a wooden spoon to stir the ingredients until a soft dough is formed. Turn the dough over onto a floured surface and knead for about ten minutes, until dough is smooth and elastic.
Shape dough into a ball and place in a bowl. Cover the bowl and let the dough rise for 1 and 1/2 to 2 hours in a warm place, until doubled in size.
Lightly knead the risen dough just to deflate it. Using a rolling pin, roll the dough into 1/2 inch thickness. Use a donut cutter and cut 16 circles from the dough. Place each circle on pre-cut individual square wax or parchment paper and place them on baking sheets. Place the smaller donut holes in there, too.
Turn the oven on for one minute only. Turn off oven and place the baking sheets inside. Fill a glass jug or bowl with boiling water and place it in the oven, too. This creates steam inside to prevent the dough from forming a skin. Let the dough rise for about 30-40 minutes, or until the size is least doubled.
Prepare your deep fryer or heat about 3 cups of cooking oil in a small, heavy bottom saucepan. Once the temperature reaches 350 F, carefully lower each donut into the oil, using the wax paper to get as low as you can without splashing. Place a few donuts at a time so they are not crowded.
Once the underside is golden, turn it over and cook the other side. Each donut should only take about 2 minutes or so to cook. The donut holes are done in a minute or less. Place each cooked donut on a tray or plate lined with paper towel.