This is my version of a dessert that is popular all over North Dakota, South Dakota, and Minnesota. It is simply referred to as "kuchen," and while kuchen translates to "cake" in German, the specific style that is popular here refers to a yeasted dough topped with custard and usually (but not always) fruit. Around here, you can get it in grocery stores and farmers' markets, and it's often served at parties. I love it with a hot cup of coffee.
For the dough: Warm the milk until just warm to the touch (90 to 100 degrees F). Stir in the yeast and a pinch of the granulated sugar and let sit until foamy, about 5 minutes. Combine the flour, remaining granulated sugar and the kosher salt in a stand mixer fitted with the paddle attachment. Mix on low speed to combine. Separate 1 of the eggs and set aside the white for the egg wash. Whisk the egg yolk and remaining whole egg into the yeast mixture. Add the yeast mixture to the flour, then mix on low speed to make a smooth, thick dough. Increase the speed to medium and add the butter a piece at a time. Once all of the butter has been added, beat on high until smooth, about 1 minute. Switch to the dough hook and knead the dough until it forms a smooth, springy ball on the hook, 4 to 5 minutes (add a little more flour, if needed, if the dough doesn't form a ball). Grease a large bowl and turn the dough in the bowl to coat. Cover and let rise at room temperature until doubled, 1 to 2 hours.
For the custard: Meanwhile, in a large saucepan, combine the cream, granulated sugar, cornstarch, kosher salt, egg and yolk. Whisk it together and heat over medium heat until thick, 4 to 5 minutes. Remove from the heat and add the butter and extract; stir until combined. Transfer to a clean bowl. Cover the surface of the custard with plastic wrap. Let cool at room temp.
Line 2 baking sheets with parchment paper. Cut the dough in half, then cut each half into 6 equal pieces. Working with one piece at a time, form the dough into balls by folding the dough under itself and pinching to seal underneath. Set the ball, smooth-side up, on a floured surface, and turn in your hand to form a tight round shape. Divide the 12 buns among the lined baking sheets. Cover loosely with plastic wrap and let rise until doubled, about 30 minutes. Preheat the oven to 350 degrees F.
Once the buns have risen, gently press down on each to make indentations that will make space for the custard. Fill the indentations with the custard. Top with the plums. Brush the buns with the reserved egg white mixed with a splash of water and sprinkle with demerera sugar, a pinch of cinnamon and flakey salt. Bake, rotating the trays from top to bottom halfway through, until the buns are risen and golden and the custard is set, 18 to 22 minutes. Transfer to a wire rack to cool.