Add sugar to the lukewarm water in a large bowl, and then add the sourdough starter. Allow it to sit until the sourdough starter starts to slightly bubble, 10 minutes. In a small bowl, beat the eggs. Cut the butter on small cubes.
Add flour, eggs, rum and lemon zest to the sourdough mixture in a large bowl. Knead well. After about 5 minutes, add the sea salt. Slowly incorporate the butter cubes and knead well with your hands. The dough has to be glossy and smooth.
Cover the bowl with cling film and leave to prove on a room temperature for about 5 hours, until it has doubled in size. Punch down the dough, then re-cover the bowl and put into the fridge to chill for 18 - 24 hours or overnight.
Turn the dough out onto a lightly floured surface and form into a rectangle. Roll the dough in one direction only, to get a 2 cm / 0.8 inch thick rectangle. Cut out 10 doughnuts with a 7.5 cm / 3 inch and a 4 cm / 1.5 inch ring mold, dip the the molds in flour before every cut. Place the doughnuts on a baking tray, lined with parchment paper and dusted with flour. Leave plenty of room between them to prove. Cover lightly with cling film and leave for about 2 hours, or until about doubled in size.
Get a heavy-based saucepan. Fill it up to the halfway point with rapeseed oil. Heat the oil to 165 - 170°C/330 - 340°F. Carefully remove the doughnuts from the baking tray, take care not to deflate them, and put them into the oil. Cover with a lid. Fry for 3 minutes on each side until golden brown. Don't cover with a lid when you turn the doughnuts around. Remove from the oil and place on kitchen paper. Set aside to cool before filling.
glaze and serve
Combine the milk and sugar in a medium mixing bowl. Whisk the mixture until completely smooth. Dip the top of the doughnuts at room temperature into the mixture at room temperature. Transfer the doughnuts on a wire rack, glaze should run down enough to cover the last quarter of the donut. Sprinkle doughnuts with sprinkles, almonds or crystalized sugar.