For choux pastry: In a pot, combine water, milk, butter, salt, and sugar, bring to a boil, and add flour. With constant stirring, evaporate for about 1–2 minutes. In a mixer with a paddle, mix until cool. Incorporate the eggs. Optionally, adjust the mixture with a bit of water. Using a piping bag with a large star nozzle, pipe onto a baking sheet lined with silicone mats choux with a diameter of 7 cm. Place in a preheated chamber on the specified program. After baking, remove from the oven and let cool. Egg yolk pastry cream: Mix 100 ml of milk with the pudding powder and yolks until smooth. Bring the remaining milk, butter, sugar, and vanilla to a boil. Pour in the mixed ingredients and cook gently for 2–3 minutes. Let cool covered with film. Egg yolk-butter cream: Beat the pastry cream until smooth with rum and incorporate softened butter. Fill a piping bag with a star nozzle with the cream. Icing: In one bowl, place sugar, lemon juice, and oil. Mix by hand until smooth into a thick icing. The icing must drip slowly from the spoon and can be well stirred. Finishing: Cool choux are sliced lengthwise, filled with cream. The tops are dipped in the icing, allowed to drain, and placed on the prepared filling.