In a large mixing bowl, combine 4 cups of all-purpose flour, 2 teaspoons of instant yeast, 1/4 cup of white granulated sugar, and 1 teaspoon of salt. Add 1 beaten egg, 1 cup of warm milk, and 1 tablespoon of vanilla extract. Mix until a sticky dough forms, then knead on a floured surface for about 8-10 minutes until smooth and elastic.

Transfer the kneaded dough to a lightly greased bowl. Cover it with a clean kitchen towel or plastic wrap and let it rise in a warm spot until it has doubled in size, approximately 60 minutes.

In a saucepan, heat 2 cups of milk over medium heat until warm. In a separate bowl, whisk together 4 egg yolks, 1/2 cup of sugar, and 1/4 cup of cornstarch. Gradually whisk this mixture into the warm milk, cooking for about 5-7 minutes until thickened. Remove from heat, stir in 1 teaspoon of vanilla extract, and let it cool completely.

Once the dough has risen, punch it down to release air and divide it into 7 equal pieces. Roll each piece into a smooth ball and place them on a floured surface. Cover with a towel and let them rise again for about 20-40 minutes until puffy and slightly larger.

In a deep pot, heat 2-3 inches of vegetable oil to 325-350°F. Carefully add the risen donuts to the oil, frying for about 2 minutes on each side until golden brown. Use a slotted spoon to transfer them to a paper towel-lined plate to drain excess oil.

Once the custard has cooled, whip 1 cup of heavy cream until it forms soft peaks. Gently fold in 1/4 cup of powdered sugar, then combine with the custard mixture. Spoon the filling into a piping bag fitted with a long tip.

Dust the cooled donuts with powdered sugar. Carefully poke a hole in the side of each donut or at the top. Pipe the creamy Bavarian filling into each donut until slightly bulging.