First, proof the yeast. Pour a teaspoon of sugar into warm milk, add active dry yeast, and stir. Let the yeast activate in a warm place; that means the yeast should be foamy on the surface. It all takes 10–15 minutes.
In the mixing bowl, pour in the flour, remaining sugar, yolk, and the activated yeast mixture with milk. Start the kitchen stand mixer at medium speed for about two minutes to roughly combine the ingredients. Then add the softened butter and salt, continuing to knead the dough—for about ten minutes—to achieve a very smooth and elastic texture.
Remove the finished dough from the dough hook, shape it into a ball, and place it in a bowl. Cover it with a clean kitchen towel or plastic wrap, and let it rise in a warm spot until it doubles in size. This usually takes about an hour to two, depending on the surrounding conditions.
Poppy seed filling: While the dough is rising, prepare the poppy seed filling. In a saucepan, heat milk with sugar; add the poppy seeds, cinnamon, lemon zest, and vanilla. Stir everything together. Reduce the heat and simmer for about ten minutes. Finally, add the rum. Let the filling cool on the kitchen counter.
Turn out the raised dough onto a lightly floured surface. Roll the dough into a roughly rectangular shape, if possible. The size depends on the baking sheet—the roll should comfortably fit lengthwise on the sheet.
Spread the chilled poppy seed filling onto the rolled-out dough, leaving about an inch of space along the edges. Fold the shorter sides over the filling first, then roll the longer side towards the other edge.
Line a baking sheet with parchment paper. Place the rolled strudel on it seam side down and let it rise in a warm place for another half an hour (the so-called second rise).
Brush the raised roll with the reserved egg white from when you prepared the dough. Use a fork to prick the surface of the roll to prevent it from cracking.
Preheat the oven to 350°F (170°C). Bake the roll for 30 minutes until the surface turns golden brown.