The next day, take 4 1/4 oz (120 g) of the activated starter and mix it with the water. Add the eggs and melted butter. Then add the sifted flours, salt, and sugar. Knead the dough. Cover and let it rise for 2 hours.

After 2 hours, flatten the dough with your fingers and shape it into a log. Place it in a basket and refrigerate for 7 hours.

For the pastry cream, heat the milk. In a bowl, mix the flour with the sugar, then add the egg yolks and vanilla extract. Mix well, then gradually add the hot milk while stirring. Once the mixture is smooth, pour it back into the saucepan and cook over heat until it thickens, which should take a couple of minutes. Stir constantly with a whisk while it thickens.

Remove the cream from the heat, transfer it to a bowl, and add the softened butter. Mix well. Cover with plastic wrap and let it cool. Once cooled, transfer the cream to a piping bag.

Take the dough out of the refrigerator, flatten it to about 1/2 inch (1 cm) thick, and cut out circles with a 2-inch (5 cm) round cutter. Cover the cut dough circles with plastic wrap and let them rest for 2 hours.

Heat the oil and fry the donuts on both sides. They fry quickly. Remove them and place on paper towels. Once cooled, fill them with the pastry cream. Dust with powdered sugar.