Put the water, buttermilk, beaten egg, and melted butter in the pan of the bread machine. Then, add the dry ingredients, except for the yeast, which was weighed with a scale. Make a small hole in the dry ingredients, and then pour in the yeast. Use the “dough” setting on the bread machine.
FOR STAND MIXER WITH DOUGH HOOK, OR BY HAND:
The dry ingredients should be put in a bowl. Put 6 tablespoons of lukewarm water in a cup and sprinkle the yeast on top. Leave it alone for about 5 minutes, until the yeast starts to react. You can help it along by adding a pinch of sugar. Make a well in the middle of the dry ingredients and pour in the buttermilk, egg, melted butter, and yeast mixture. If you are using a stand mixer, use the dough hook and run the mixer until a dough forms, then keep going for about 5 minutes. Knead the dough for about 10 minutes by hand. Cover and let it sit for at least two hours, depending on how fresh the yeast is, the temperature of the water, and the temperature of the room.
BOTH BREAD MACHINE, STAND MIXER/HAND INSTRUCTIONS FOLLOW BELOW:
When the dough has doubled in size, put it on a surface dusted with flour and knead it lightly. Cut it in half and keep half of the dough covered so that it doesn’t get skin. Roll out half of the dough to about 1/2-inch thick with a rolling pin. Cut with a sharp, round cookie cutter with a diameter of about 3 inches. Then, use a smaller cookie cutter with a diameter of about 1 inch to make holes and save the holes. Or you could buy this doughnut cutter and save yourself a lot of trouble.
Put each doughnut on a piece of parchment or waxed paper and then on a cookie sheet. Put the tray in the oven (turn it on for 1 minute, SET A TIMER, and then turn it off again, just to make it barely warm).
Next, bring some water to a boil and pour it into a measuring jug. Put the jug of water and the tray of doughnuts in the oven. This will make steam, which will keep skin from forming. Cut the rest of the dough into quarters, and then cut each quarter in half to make 8 equal pieces. Roll each piece of dough into a smooth ball and place it on a piece of parchment paper or waxed paper on a cookie sheet. Put the cookie sheet in the oven with the other doughnuts to rise until they are twice as big.
The oil should be heated to about 350oF (180o). If you don’t have a thermometer, use a doughnut hole to test the temperature of the oil. If the hole doesn’t start to fry right away, the oil is too cold. If the hole turns brown right away, the oil is too hot. Change the heat as needed.
You can use the paper to carefully lower the doughnuts into the hot oil. When the bottoms are golden brown, flip them over. When they’re done, take them out and put them on a paper towel-lined plate.
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