1. Activate the yeast by mixing dry active yeast, warm milk, and sugar, then let it sit for 5-10 minutes until foamy.
2. Create the dough base by combining all-purpose flour, salt, and the activated yeast mixture in a mixing bowl.
3. Incorporate wet ingredients, including eggs, melted butter, and optional vanilla extract, into the flour mixture.
4. Knead the dough until it becomes smooth and elastic, either by hand or using a stand mixer for about 8-10 minutes.
5. Let the dough rise in a lightly greased bowl, covered with a towel, in a warm, draft-free area until doubled in size (approximately 1-2 hours).
6. Shape the doughnuts by rolling the dough to ½ inch thickness on a floured surface and cutting out rounds using a cookie cutter.
7. Allow the shaped doughnuts to rise again on a baking sheet covered with a towel for an additional 45 minutes to an hour.
8. Heat vegetable oil in a fryer or deep pan to 350°F (175°C) and fry the doughnuts for 2-3 minutes on each side until golden brown.
9. Remove the doughnuts from the oil using a slotted spoon and drain them on paper towels.
10. While warm, roll the fried doughnuts in sugar to coat them evenly.
11. Fill the doughnuts with jam using a pastry bag fitted with a donut tip, carefully inserting and filling each donut.
12. Serve the jam donuts warm or store them using proper techniques to maintain freshness for later consumption.