Heat the milk to 110° and transfer it to a small bowl. Stir in the yeast and let stand for 5 minutes, until dissolved. Stir in the egg. In a large bowl of a standing electric mixer fitted with a dough hook, mix the 4 cups of flour with the sugar and salt; add the warm milk mixture and knead at medium-low speed until a stiff dough forms, about 3 minutes. Add the softened butter and knead at medium speed until the dough is silky and soft, about 5 minutes. Transfer the dough to a buttered bowl, cover with plastic wrap and refrigerate overnight.
In a medium bowl, combine the flour with 1/4 cup of the sugar and 1/4 cup of the milk and whisk to a paste. Whisk in the eggs and vanilla. In a medium saucepan, heat the remaining 1 1/4 cups of milk and 1/4 cup of sugar over moderate heat until the sugar is dissolved. Gradually whisk the hot milk into the egg mixture, then cook the mixture in the saucepan over moderate heat, whisking constantly, until thickened and no floury taste remains, about 4 minutes. Transfer the pastry cream to a bowl and cover with plastic wrap, pressing it directly onto the surface. Refrigerate overnight.
Butter four 9-inch pie plates. Divide the dough into 4 pieces and flatten into disks (it will be a firm dough). On a lightly floured work surface, roll out each disk of dough to a 9 1/2-inch round. Place each round in a pie plate, making sure that the dough doesn't extend past the rims. Cover the pie plates with plastic wrap and let stand at room temperature until the dough has risen slightly, about 1 hour.
Preheat the oven to 350° and set racks in the upper and lower thirds. Lightly press the dough with your fingers to deflate it. Arrange the apricots on the bottom in a single layer. Spread the pastry cream evenly over the apricots.
In a medium bowl, combine the sugar with the flour and butter and rub with your fingers until the texture is sandy. Sprinkle the topping over the pastry cream and dust lightly with cinnamon. Bake the kuchens for 40 minutes, until the crusts are deeply golden. Transfer the kuchens to a rack to cool completely before serving.