Place butter in the mixing bowl of a stand mixer. Pour hot milk over butter; stir occasionally until butter melts. Whisk pumpkin puree, 3/4 cup white sugar, eggs, and egg yolk into milk mixture.

In a separate bowl, combine flour, yeast, salt, ginger, cinnamon, and allspice. Using the paddle attachment, mix in about 4 cups of the flour mixture on low speed. Scrape down the sides of the bowl and underneath paddle. Add the remaining flour and mix until fully combined. The dough will be wet.

Transfer the dough to a greased bowl. Cover with plastic wrap and refrigerate for at least 5 hours or overnight.

Combine cranberry sauce with 1/3 cup water in a small saucepan. Break the sauce up slightly with a spoon. Heat over medium-low heat, stirring occasionally, until the sauce melts. Pour into a bowl and refrigerate.

Place chilled dough on a floured surface, shape into a ball, and divide in half. Store half the dough in the fridge. Roll out the remaining dough to 1/4-inch thickness. Cut out rounds with a 3-inch cutter.

Heat oil to 375°F (190°C) in a deep-fryer or saucepan. Line a baking sheet with a non-stick liner. Arrange dough rounds on the sheet in pairs.

Whisk egg white with water. Brush the bottom halves of doughnut pairs with the mixture. Spoon cranberry jelly in the center and cover with the top half, pressing edges to seal.