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In a small saucepan, heat milk until it is warm, about 90 degrees. In a large bowl, combine the warmed milk and yeast. Let sit until the mixture is foamy, about 5 minutes.
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Beat the eggs, butter, salt, and sugar into the yeast mixture. Add in half of the four and mix until combined.
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Mix in the rest of the flour. When the mixture is too thick to beat, transfer to a floured surface and gently knead it until smooth.
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Grease a large bowl with a little oil. Transfer the dough to the bowl and cover with plastic wrap or a clean kitchen towel. Let rise until it doubles in size, about 1 hour.
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Place the dough onto a floured surface and roll it to ½ inch thickness.
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Cut out six round doughnuts using a 3-inch round cookie cutter and six rectangle donuts using a sharp knife.
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Place the cut out doughnuts on parchment lined baking sheets. Cover with a clean kitchen towel and place in a warm place until they are slightly puffed and delicate, about 45 minutes.
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About 15 minutes before the doughnuts are done rising, pour the oil into a heavy bottomed pot or dutch oven over medium heat. Heat the oil to 375 Line wire cooling racks with paper towels.
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Carefully add doughnuts to the oil, a couple at a time. When the bottoms are golden, about 45 seconds, flip and cook until they’re golden all around.
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Use a slotted spoon to remove doughnuts from the oil and place on the prepared cooling racks and let cool.