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In a large mixing bowl, proof yeast in warm water for about ten minutes, until frothy.
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While yeast is proofing, add milk and butter to a small saucepan and heat over medium heat until milk is slightly scalded (will look bubbly) on the bottom.
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Remove from heat and add sugar, hot pumpkin, and cinnamon. Mix and set aside to cool to lukewarm.
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Add egg, salt, and 1 cup of flour to the yeast mixture and beat well.
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Add the pumpkin/milk mixture and beat well.
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If using an electric mixer, switch to a dough hook for this step. Add 2 1/2 to 3 cups of flour and mix until a slightly sticky dough is formed.The dough should barely stick to your fingers.
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Cover and let rise until doubled in size.
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Punch down and roll out 1/2" thick onto a heavily floured surface.
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Use a large cup to cut into doughnuts and a small medicine cup to cut out the holes.
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Rest doughnuts and holes for 30 minutes.
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Heat oil (at least 2 inches) to 375 degrees.
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Carefully place doughnuts and holes into oil in batches, frying and flipping so that both sides brown evenly.
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Remove from oil and set on paper towel to cool.
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When cool, drizzle with maple glaze.