Add the lukewarm milk, yeast, and granulated sugar to the bowl of a stand mixer fitted with the dough hook. Stir briefly to moisten the yeast, then add the flour, salt, softened butter, egg yolk, vanilla extract, and lemon zest.
½ cup (120 ml) whole milk, 2¼ tsps (7 g / 1 packet) active dry yeast, ¼ cup (50 g) granulated sugar, 3 cups (375 g) all-purpose flour, ½ tsp (3 g) table salt, 1 large egg yolk , 1 tsp (4 g) vanilla extract, zest of one lemon, ½ cup (113 g) unsalted butter
Mix on low speed until a shaggy dough forms and no dry flour remains. Increase to medium-low speed and knead for about 5 minutes, until the dough is smooth, soft, and slightly tacky. Avoid adding extra flour unless the dough remains excessively sticky after kneading.
Shape the dough into a ball and return it to the bowl. Cover and let rise in a warm, draft-free place for about 1 hour, or until approximately doubled in size.
While the dough rises, wash and thoroughly dry the Italian prune plums. Cut the plums in half lengthwise and remove the pits, then cut each half lengthwise into 4 narrow pieces so each whole plum yields about 8 pieces. Set aside.
1-1¼ lbs (454-570 g) Italian prune plums
To make the streusel, stir together the flour, granulated sugar, cinnamon, and salt. Add the melted butter and vanilla extract and mix with a fork until crumbs form, leaving a mixture of smaller crumbs and larger pieces. Set aside.
½ cup (113 g) unsalted butter, 1½ cups (190 g) all-purpose flour, ¾ cup (150 g) granulated sugar, ½ tsp (1 g) ground cinnamon, pinch table salt, 1 tsp (5 ml) vanilla extract
Butter a 9 x 13-inch (23 x 33 cm) sheet pan. Line the bottom with parchment paper if desired.
Transfer the risen dough to the prepared sheet pan and use lightly buttered fingertips to press and stretch it into an even layer across the bottom. Cover and let rise in a warm, draft-free place for about 30 minutes, until the thin layer of dough is noticeably puffier and approximately doubled in thickness.
While the dough completes its second rise, preheat the oven to 375°F (190°C).
Arrange the plum pieces over the risen dough in 15 groups, corresponding to the 15 portions of the finished cake. Alternate the direction of the plum pieces from one group to the next to create the cross-hatch pattern.
Scatter the streusel evenly over the plums and exposed dough, keeping some of the larger crumbs intact.
Bake for 30 to 40 minutes, rotating the sheet pan halfway through, until the dough is golden brown around the perimeter, the streusel is lightly browned, and the plums are tender with their juices bubbling.
Cool the German plum cake in the pan for at least 20 minutes before cutting into 15 pieces, using the arrangement of the plums as a guide. Serve warm or at room temperature.