Add flour and sugar to a bowl of a stand mixer fitted with a dough hook. Feel free to use a hand-held electric mixer fitted with two dough hooks. Make a well in the middle. Pour lukewarm milk and lukewarm water into the well. Add the active dry yeast and cover the ingredients with some flour. Add the eggs, egg yolk, grated lemon zest, rum, and salt. Start kneading on low, set the stand mixer to medium speed, and knead for about 6 - 8 minutes or until the dough is elastic, smooth, and easily separates from the bowl. Add the butter and knead for another 3 - 4 minutes for the butter to incorporate into the dough and separates from the bowl easily.TipThe water and milk temperature should be around 30 - 35°C or 85 - 95°F.If you're using all-purpose flour instead of Manitoba flour or bread flour, use 20g less milk (2 tbsp).
Cover the bowl with the dough with clingfilm and leave to rise at room temperature (20 - 25°C / 68 - 77°F) until visibly increased, for about 1 hour - 1 1/2 hours.
Lightly dust a large baking sheet with flour. Divide the dough into 14 pieces. Each piece should weigh around 70g / 2.5 ounces. Shape each piece into a smooth ball. Place the balls seam-side down onto the prepared baking sheet and repeat the process until all the balls are shaped. Leave enough space between the doughnuts to rise. Cover with a kitchen towel. Lightly sprinkle the towel with water. Leave the doughnuts to rise at room temperature for about 1 1/2 hours - 2 1/2 hours or until they triple in size.
Place a large pan (with a lid) over medium heat. Add the oil and wait until the oil reaches 170 - 180°C / 340 - 355 °F. Lightly dust your spatula with flour and carefully pick one doughnut. Turn it around in your hands so the seam looks up, and gently drop it into the oil. Add about 4 - 5 more doughnuts, depending on the size of the pan. The doughnuts should have some space to move, but not too much. Shake the pan slightly to distribute the doughnuts. Cover with a lid and fry for 2 - 3 minutes, then remove the lid, turn the doughnuts, and fry for another 2 - 3 minutes (uncovered). Transfer the doughnuts onto a paper towel-lined wire rack. Repeat the process until you fry all the doughnuts.
Berliner Doughnut with Jam
Add the apricot jam to a saucepan and place over low heat to warm it. Transfer to a pastry bag and pipe the doughnuts with the jam while they are still warm. Place on a serving plate and dust with powdered sugar.