Activate Yeast: In a bowl, mix warm milk and sugar, then sprinkle the yeast over. Let it sit for about 5 minutes until frothy.

Mix Ingredients: In a large mixing bowl, combine flour, salt, egg, and the frothy yeast mixture. Mix until well combined.

Add Butter: Add the softened butter and knead the dough for 8-10 minutes until smooth and elastic.

Let It Rise: Cover the bowl with a cloth and let the dough rise in a warm place for 1-1.5 hours, or until it doubles in size.

Whisk Ingredients: In a saucepan, whisk together the egg, yolk, sugar, flour (or cornstarch), and salt until smooth.

Heat Milk: Heat the milk until warm but not boiling. Gradually pour the warm milk into the egg mixture, whisking continuously.

Cook the Custard: Place the mixture over medium heat and cook, whisking constantly, until thickened. Remove from heat and stir in the vanilla extract. Let it cool completely.

Roll the Dough: After the dough has risen, punch it down and roll it out on a floured surface to about ½-inch thickness.

Cut the Donuts: Use a donut cutter or a round cookie cutter (about 3 inches) to cut out donuts. Place them on a baking sheet lined with parchment paper. Cover with a cloth and let rise for another 30 minutes.

Heat Oil: Heat oil in a deep fryer or large pot to 350°F (175°C).

Fry Donuts: Fry the donuts in batches for about 1-2 minutes per side until golden brown. Drain on paper towels.

Fill the Custard: Transfer the cooled custard into a piping bag fitted with a round tip. Poke a hole into the side of each donut and fill it generously with the custard.

Dust with powdered sugar or glaze as desired and enjoy warm!