Heat the butter, milk, water, and sugar in a pot over medium heat and bring to a simmer. Add the flour and stir the mixture with a wooden spoon until a smooth dough forms. Remove the pot from the heat, let it cool a bit, and then mix in the eggs one at a time.
Line 3x 9-inch springform cake pans with parchment paper and grease the sides. Divide the choux pastry between the 3 pans and spread out the dough. Transfer all 3 pans to a preheated oven and cook at 350°F for 30 minutes. Make sure not to open the oven door when the cakes are inside, otherwise the choux pastry will collapse.
Add the sugar, eggs, corn starch, vanilla, and milk to a saucepan over medium heat and bring the mixture to a boil while whisking constantly. Cover the pot with plastic wrap and refrigerate until the vanilla pudding reaches room temperature. Beat the butter with a hand mixer until fluffy and then gradually mix in the chilled vanilla pudding until a smooth cream forms.
Place one of the cooled choux pastry cakes on a cake plate and put the ring of a 9-inch springform cake pan around it. Spread 1/4 of the vanilla pudding over the pastry cake. Flip the second choux pastry cake over and spread 1/4 of the vanilla pudding over the bottom of it. Turn the cake right side-up so that the vanilla pudding side is facing down, and place it on top of the vanilla pudding already inside the springform ring. Spread 1/4 of the vanilla pudding over the top of the second choux pastry cake in the springform ring. Flip the third choux pastry cake over and spread 1/4 of the vanilla pudding over the bottom of it. Turn the cake right side-up so that the vanilla pudding side is facing down, and place it on top of the vanilla pudding already inside the springform ring. Refrigerate the karpatka cake for 4-5 hours, or preferably overnight. Remove the springform ring and sprinkle powdered sugar on top before serving.