In a small bowl, sprinkle the yeast over the lukewarm milk. Stir briefly and allow to sit at room temperature until frothy, about 10 minutes.
In the bowl of a stand mixer fitted with a dough hook or a large bowl, combine the flour, sugar, lemon zest, and salt.
Mix in the milk with frothy yeast, egg yolks, egg, and butter until a soft dough forms.
Knead on a lightly floured surface just until elastic and smooth. Place in a large bowl, cover, and allow to rest at room temperature until doubled, 1-2 hours.
On a lightly floured surface, roll the dough into a sheet about 3/4 inch (2 centimeters) thick.
Cut out circles (do not twist) with a 3 inch (7.5 centimeter) circular cutter. Bring together the scraps of dough and repeat to get as many doughnuts as possible.
Place the doughnuts 2 inches (5 centimeters) apart, cover with a towel, and allow to rise at room temperature until puffy, 30-60 minutes.
Pour 2 inches (5 centimeters) of vegetable oil in a large saucepan and heat to 340˚F (170˚C), no higher.
Once heated, gently add a few of the puffed doughnuts, taking care not to overcrowd. Immediately cover with a lid.
Fry until lightly golden on the bottom, about 2 minutes, then remove the lid and flip to fry uncovered until golden on the other side.
Transfer to a towel-lined platter and repeat with remaining doughnuts, keeping the temperature no higher than 340˚F (170˚C).
If coating with granulated sugar, toss the doughnuts in a bowl of the sugar immediately after removing from the oil. If coating with powdered sugar, wait until after piping the doughnuts to coat.
Once the doughnuts have slightly cooled, fill a pastry bag with a Bismark piping tip with desired jam.
Pipe the jam into each doughnut, then top with powdered sugar.