Combine lukewarm milk, 1 tablespoon of sugar, and yeast in a bowl. Let it sit in a warm place for 5–10 minutes until it becomes frothy.
In a large bowl, combine flour, remaining sugar, salt, and lemon zest. Add the yeast mixture, softened butter, eggs, and vanilla extract. Mix until a dough forms.
Knead for 8–10 minutes by hand or with a stand mixer with a dough hook attachment until smooth. The dough should pull away from the sides of the bowl but remain soft.
Place the dough in a greased bowl, cover with a kitchen towel, and let the dough rise in a warm place for about 1 hour or until doubled in size.
Punch down the dough. Weigh the dough, divide the total weight by 12, then cut and adjust each piece to that weight. Roll each piece into smooth balls and place them on a parchment-lined baking sheet. Cover and let them rise for another 30 minutes until puffy.
Preheat your oven to 375°F (190°C). Bake the dough balls for 12–15 minutes, or until lightly golden. Remove from the oven and transfer to a cooling rack.
Once cooled slightly, use a piping bag with a filling tip to inject your jam into the side of each donut. Be careful not to overfill.
Brush the donuts with melted butter while still warm. Roll them in granulated sugar for Krapfen or dust them with powdered sugar for Berliners.
