Making the dough: Sprinkle yeast over lukewarm milk and let sit for 5-10 minutes, until slightly foamy.
In the bowl of a standing mixer, mix flour and sugar. Fit mixer with dough hook and add egg, butter, zest, vanilla extract, brandy, and milk mixture to the flour mixture. Mix on low speed until a soft ball of dough starts to form, 2-3 minutes. If the dough is too sticky or soft and doesn’t come together, add more flour as necessary, one tablespoon at a time. Add in salt. Keep mixing until dough is soft and elastic and pulls away from the sides of the bowl, about 8 minutes. Place dough in a lightly oiled bowl and cover with plastic wrap. Let rest in a warm place for about 2 hours, or until doubled in size.
When the dough has risen to twice its original size, gently punch it to remove air. Afterward, divide it into 24 pieces (or as many pieces as you like, depending on your preferred size). The best way to do this is to divide the dough into two halves, shape each half into a log, and cut each log into 12 equal pieces. Roll each piece into a ball and place on a lightly oiled baking sheet. Cover loosely with plastic wrap and let rest to rise for about 60 minutes, or until doubled in size.
Frying: In a large, deep, heavy-bottomed saucepan, heat 2 in./5 cm of oil until a thermometer inserted into the oil reaches 300F/150C. If you don’t have a thermometer to test, insert a wooden spoon into the oil; it’s ready for frying when bubbles appear around the stick. If it bubbles vigorously, it’s too hot. Carefully add doughnuts, a few at a time, depending on how large the saucepan is. Fry until golden brown, about 2 minutes on each side. Drain doughnuts and move to paper towel-lined plate.
Filling with jam: Once doughnuts have cooled, place jam in a piping bag fitted with a 1cm round tip opening. Press tip halfway into doughnuts and squeeze until jam begins to dollop out. Top doughnuts with powdered sugar.
Doughnuts are best the same day they are made.