Pour ¼ cup of the milk, warmed to lukewarm (about 100°F), into a glass. Stir in ½ teaspoon of the sugar and the yeast. Rest in a warm spot 10–15 minutes, until foamy.
Melt 5 Tablespoons unsalted butter, take it off the heat, and pour in the remaining milk to cool it. Whisk in 1 egg. The mixture should be lukewarm, not hot.
In a large bowl, stir together 2 cups all-purpose flour, the remaining sugar, and ½ teaspoon salt. Make a well, pour in the butter mixture and the foamy yeast, and combine roughly with a fork.
Knead with the dough hook (lowest speed 1–2 minutes, then medium-low about 8 minutes), with a wooden spoon until the dough blisters and leaves the spoon, or by hand about 10 minutes. The dough is smooth, elastic, and only slightly sticky.
Cover and let the dough rise in a warm, draft-free spot for 1½ to 2 hours, until doubled. Toward the end of the rise, knead it lightly for a moment and let it rise again.
Meanwhile, rub the streusel 5 Tablespoons unsalted butter, ¾ cup all-purpose flour, and 6 Tablespoons granulated sugar into moist crumbs and chill them.
Rub the pan lightly with butter and line it with parchment paper.
Preheat the oven to 350 °F. Press or roll the dough onto the pan into a thin, even layer with a slightly raised rim.
Arrange the plum quarters cut side up, close together over the whole surface. Sprinkle with the ground poppy seed, if using, and with the sugar if the plums taste tart. Scatter the chilled streusel evenly on top.
Bake for about 30 minutes, until the edges turn golden.
Melt the drizzle 5 Tablespoons unsalted butter, stir in 1 ½ Tablespoons rum, and spoon the mixture over the hot cake as soon as it comes out of the oven.
Let the cake cool in the pan, then cut it into 15 squares (3 rows by 5).