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1
Let cider stand until it is at room temperature. Melt butter with syrup in a small saucepan over low heat; cool slightly. Sift dry ingredients together. Beat eggs in a separate bowl. Add the butter/syrup mixture, cider, and vanilla. Beat in pureed squash. Fold in the dry ingredients, about 1/4 at a time, stirring after each addition until finished. Cover with plastic; chill 3 hours.
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2
Sprinkle 2 rimmed baking sheets lightly with flour. Press out 1/3 of dough on floured surface to about 1/2-inch thickness. Add a little extra flour to the dough if necessary (the dough should be firm enough to roll while keeping the dough soft). Shape with doughnut cutter or drinking glass (about 2 1/2-inch diameter), cut out dough rounds then arrange on sheets. Repeat with remaining dough in 2 more batches. Gather dough scraps. Press out dough and cut out more dough rounds until all dough is used.
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3
Using 1-inch diameter round cutter, cut out center dough round to make doughnuts and doughnut holes.
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4
Line 2 baking sheets with several layers of paper towels. Preheat the oil in a Dutch oven 365F to 370F (use a deep fry thermometer). Fry doughnut holes in 2 batches until golden brown, turning occasionally, about 2 minutes. Using slotted spoon, transfer to paper towels to drain. Fry doughnuts, 3 or 4 at a time, until golden brown, adjusting heat to maintain temperature, about 1 minute per side or until golden brown. Using slotted spoon, transfer doughnuts to paper towels to drain. Cool completely.
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5
Combine sugar and cider in a small saucepan over medium heat. Stir until sugar is dissolved, and bring to boiling point. Reduce heat and simmer for five minutes, stirring constantly. Remove from heat.
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6
Arrange doughnuts on racks. Pour glaze over doughnuts on 1 side. Allow the glaze to set, about 30 minutes or so.
