Step 1: In the bowl of a stand mixer, combine the warm milk, yeast, and 1 tablespoon of the granulated sugar. Stir gently and let it sit for 5-10 minutes until foamy. This indicates the yeast is active.
Step 2: Add the remaining granulated sugar, eggs, and vanilla extract to the yeast mixture. Mix well with a whisk.
Step 3: Add the bread flour and salt to the wet ingredients. Attach the dough hook to your stand mixer and mix on low speed for 2 minutes until a shaggy dough forms. Scrape down the sides of the bowl.
Step 4: Increase the speed to medium-low and knead for 5 minutes. The dough will be sticky. Begin adding the softened butter, one piece at a time, allowing each piece to fully incorporate before adding the next. This process should take about 5-7 minutes.
Step 5: Continue kneading on medium speed for another 10-15 minutes until the dough is smooth, elastic, and pulls away cleanly from the sides of the bowl. It should pass the ‘windowpane test’ (you can stretch a small piece thin enough to see light through it without it tearing).
Step 6: Transfer the dough to a lightly oiled large bowl, turning it once to coat. Cover the bowl tightly with plastic wrap and let it rise in a warm place for 1.5-2 hours, or until doubled in size. Then, punch down the dough, re-cover, and refrigerate for at least 8 hours or overnight. This cold fermentation develops flavor and makes the dough easier to handle.
Step 7: The next day, remove the dough from the refrigerator. On a lightly floured surface, roll the dough to about ½-inch thickness. Using a 3-inch round cutter (or a glass), cut out 12 doughnuts. Re-roll scraps gently and cut out more if possible.
Step 8: Place the cut doughnuts on individual squares of parchment paper on a baking sheet. Cover loosely with plastic wrap or a clean kitchen towel and let them proof in a warm, draft-free place for 1-1.5 hours, or until visibly puffy and light.
Step 9: In a large heavy-bottomed pot or Dutch oven, heat the vegetable oil over medium heat until it reaches 325-350°F (160-175°C) on a deep-fry thermometer. Have a plate lined with paper towels ready.
Step 10: Carefully lift the doughnuts (still on their parchment squares) and gently drop them into the hot oil, peeling off the parchment paper. Fry 2-3 doughnuts at a time to avoid overcrowding. Fry for 1.5-2 minutes per side, until golden brown. Adjust heat as needed to maintain oil temperature.
Step 11: Using a slotted spoon or spider, transfer the fried doughnuts to the paper towel-lined plate to drain excess oil. While still warm, roll them in granulated sugar (placed in a shallow dish) until evenly coated.
Step 12: Once the doughnuts have cooled slightly but are still warm (to prevent the jam from melting), fill a piping bag fitted with a long, thin Bismarck or pastry tip with the plum jam. Insert the tip into the side of each doughnut and gently squeeze to fill. You’ll feel the doughnut plump up slightly. Serve immediately or store in an airtight container for up to 2 days.