Moist Sheet Pan Butter Cake with Pudding, Just Like from the Pastry Shop

Ingredients

3 cups380 gall-purpose flourGerman Type 405. In German recipes plain flour means soft wheat flour Type 405 unless stated otherwise. In the US use all-purpose flour; in the UK, plain flour.
½ cubeyeast½ cube = 21 g fresh yeast. German recipes use either a 42 g cube of fresh yeast or a 7 g packet of dry yeast; one cube equals about two packets. In the US use active dry or instant yeast; one ¼ oz (7 g) envelope replaces half a cube of fresh yeast. In the UK use fast action dried yeast (7 g sachet per half cube), or fresh yeast from a supermarket bakery counter.
¼ cup50 ggranulated sugar
1large eggroom temperature
3 ½ tbsp50 gunsalted buttersoftened. German butter is unsalted and has about 82 percent fat. In the US use unsalted butter; European-style unsalted butter is closest. In the UK use unsalted butter.
¾ cup plus 1 tbsp200 mlmilkwarm

3 cups700 mlmilk
6 tbsp80 ggranulated sugar
½ cup plus 2 tbsp80 gcornstarch or custard pudding powderCornstarch is a fine starch, usually made from corn, used to thicken creams and lighten cakes (US: cornstarch; UK: cornflour). German custard pudding powder is a packet of flavored starch (about 37 g) that is cooked with 2 cups (500 ml) of milk and sugar into a thick pudding. In the US use about ⅓ cup (40 g) cornstarch plus 1 tsp vanilla extract per packet; in the UK, custard powder, about 40 g per German packet (it is unsweetened, so it swaps 1:1 by weight).
4applesFuji or Cox
1 pinchcinnamon
1 splashrumoptional
2 tspvanilla sugar1 packet = 2 tsp (8 g). Vanilla sugar is sugar flavored with vanillin (artificial vanilla), sold in small packets of about 8 g. In the US use 1 tsp vanilla extract plus 2 tsp sugar for each 8 g packet; in the UK, vanilla sugar sachets, or 1 tsp vanilla extract plus 2 tsp sugar per sachet.

Make a well in the flour, add the sugar, milk & yeast and let it sit for 10 minutes until it starts to bubble, then add the butter & egg and knead everything into a dough.
Cover the dough and let it rest for about 1 hour.
In the meantime, cook the filling. To do this, whisk a splash of the milk thoroughly with the cornstarch & egg, and bring the rest of the milk to a boil.
As soon as the milk starts to boil, stir in the cornstarch mixture and, stirring constantly, let it bubble for at least 1 minute.
Transfer the pudding to another container, cover it and let it cool completely.
Peel, quarter and core the apples and cut them into cubes. Cook them in a pot with a little cinnamon & the almonds until soft, then let them cool.
Take the dough out of the bowl and place it on a large baking sheet (about 16 × 14 inches / 40 × 35 cm; a US half sheet pan, 18 × 13 inches, works too) lined with parchment paper. Roll it out to about ⅜ to ⅝ inch (1 to 1.5 cm) thick, then poke holes in it with a wooden spoon and fill them with butter.
Let the dough rest for about 30 minutes, then preheat the oven to 350°F (180°C) convection (fan) and bake until golden (about 20 minutes). If you like, sprinkle it with a little extra sugar before baking.
After baking, let the cake cool and slice it in half horizontally.
Give the pudding a quick stir, then fold in the apples. Spread it over the bottom layer and place the top layer on it.

From German Pastry Chef. Full recipe with photos: germanpastrychef.com/recipes/german-butter-cake-with-pudding/