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For the doughnuts: Cream the sugar and butter together for about 6 minutes on medium speed. Since there is a lot of sugar the mixture won't get fluffy but the butter should be soft and the sugar evenly dispersed. Add the eggs and yolks one at a time, mixing well between each addition.
Stir together the buttermilk, sour cream and reserved 1 3/4 cups plus 2 tablespoons squash puree (use alternatively canned pumpkin puree) in a medium bowl. Slowly incorporate the squash mixture into the butter mixture until fully combined.
Sift together the all-purpose flour, cake flour, baking powder, ginger, salt, baking soda, cinnamon, mace and nutmeg. Add the dry ingredients to the wet ones and mix to combine, scraping down the bowl to ensure that all of the ingredients are fully incorporated. Place the dough on plastic wrap and roll into a flat square. Wrap tightly and chill in the refrigerator for at least 2 hours or overnight.
For the candied pepitas: Preheat the oven to 275 degrees F and line a cookie sheet with parchment paper. Toss the pepitas and confectioners sugar in a bowl. Gradually add a few drops of water and mix until the sugar has dissolved and the mixture coats the pepitas evenly. Add the salt and toss to coat. Spread the pepitas on the prepared cookie sheet and bake, stirring every 5 minutes, until the sugar starts to crystalize, 15 to 20 minutes.
For the ginger glaze: Whisk together the confectioners sugar, ginger juice and 4 tablespoons of water until smooth. Store the glaze in an air-tight container in the refrigerator for up to two weeks.
For the spiced sugar: Mix together the granulated sugar, ginger, cinnamon, nutmeg and salt in a medium bowl until well combined.
To make the doughnuts: Heat 6 to 8 inches of the oil or shortening in a deep pot to 325 degrees F. Roll out the dough on a well-floured surface until it is 1/4-to-1/2-inch thick. Use 3-inch and 1-inch round cookie cutters to cut out the doughnuts.
Fry the doughnuts, 2 to 3 pieces at a time, for 1 minute and 30 seconds on each side. Check the temperature of the oil between batches. If the temperature has dropped, allow it to return to 325 degrees F before frying another batch.
While the doughnuts are hot, top them with the ginger glaze and immediately sprinkle them with the candied pepitas, pressing in as needed so they stay attached. Apply the glaze liberally so it covers the doughnuts well; scrape up the excess to use on the rest of the doughnuts.
Re-roll the dough scraps and cut out rounds with the 1-inch cookie cutter. Fry for about 2 1/2 minutes. Toss the doughnuts in the spiced sugar mixture while still hot.
Store the doughnuts for up to 1 day in a covered container.