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Sprinkle the yeast and a pinch of sugar over the warm water, and let stand for 5 minutes, or until foamy.
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In a large bowl, mix together the yeast mixture, milk, sugar, salt, eggs, butter, and 2 cups of the flour. Mix for a few minutes at low speed, scraping the bowl a few times.
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Beat in remaining flour 1/4 cup at a time, until the dough no longer sticks to the bowl. Knead for about 5 minutes, or until smooth and elastic.
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Place the dough into a greased bowl, and cover. Set in a warm place to rise until double.
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Turn the dough out onto a floured surface and divide. (You can freeze one half at this point)
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Working with one half at a time, gently roll out to 1/2 inch thickness. Cut with a floured donut cutter.
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Let donuts sit out to rise again until double. Cover loosely with a cloth or plastic wrap.
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For the glaze, melt butter in a saucepan over medium heat. Stir in confectioners' sugar and vanilla until smooth. Remove from heat, and stir in hot water one tablespoon at a time until the icing is thin enough. Set aside.
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Heat oil in a deep-fryer or large heavy skillet to 350 degrees F (175 degrees C). Gently place the donuts into the oil. I just pick each one up by hand.Turn donuts over as they rise to the surface. Fry on each side until lightly golden. Remove from hot oil to drain on a wire rack.
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Dip doughnuts into the glaze while still hot, and set onto wire racks to drain off excess. For easy clean up, place a piece of parchment under the wrack. ENJOY!