Place butter in the mixing bowl of a stand mixer. Pour hot milk over butter; stir occasionally until butter melts. Whisk pumpkin puree, 150g sugar, eggs and egg yolk into milk mixture.
In a separate bowl, combine flour, yeast, salt, ginger, cinnamon and allspice. Using the paddle attachment of your mixer, mix in about 500g of the flour mixture into the butter mixture on low speed. Scrape down sides of bowl and underneath paddle. Add remaining flour and mix, scraping down sides, on low speed until fully combined. Dough will be very wet, almost cookie-dough consistency; don't worry, it will absorb moisture as it sits.
Transfer dough to a greased bowl. Cover bowl with greased cling film. Chill in the fridge for at least 5 hours or overnight.
Add cranberry sauce to a small saucepan. Heat, covered, over medium-low heat, stirring occasionally, until cranberry sauce melts, about 25 minutes. Add water as needed to thin the sauce to a jam-like consistency. Pour into a bowl and chill in the fridge till needed.
Place chilled dough on well-floured work surface. Shape it into a ball, divide the dough in half, and return half the dough to the greased bowl. Recover with the greased cling film and place in the fridge till needed. Roll out remaining dough half, flouring surface as needed, to 5mm thickness. Cut out dough rounds with a 7cm round cutter.
Line a baking tray with greaseproof paper or a silicone liner. This will prevent the dough from sticking and tearing.
Transfer dough rounds to prepared baking tray. Gather up remaining dough scraps and add them to the dough stored in the fridge. Roll out and cut into rounds as before.
Whisk egg white with 1 teaspoon water. Brush the rounds that will be the bottom halves of the doughnuts with the beaten egg white.
Spoon 1 teaspoon cranberry sauce into centre of each bottom round; arrange second round on top and press edges to seal. Let doughnuts rest 20 minutes.
Heat oil in a deep-fryer or large saucepan to 190 degrees C.
Working in batches, fry doughnuts until well browned, about 1 1/2 minutes per side. Drain doughnuts on kitchen paper. Pour granulated sugar into a bag or large bowl; toss doughnuts in sugar to coat. Dust doughnuts with icing sugar before serving.