Make the brioche dough

• Warm the milk and combine it with the flour, sugar and instant yeast in the bowl of a stand mixer.
• Mix until a shaggy dough forms, then add the eggs, egg yolk and salt.
• Knead with the dough hook until the dough is smooth and pulls away from the bowl.

Incorporate the butter

• Add the softened butter a little at a time while kneading.
• Keep kneading until the dough is soft, elastic and passes the windowpane test.

Let the dough rise

• Shape the dough into a ball and place it in a lightly oiled bowl.
• Cover and let it rise in a warm spot until doubled in size.

Cook the apple pie filling

• Cook the diced apples with the brown sugar, cinnamon and lemon juice in a saucepan over medium heat until the apples are tender.
• Mix the cornstarch with the water and stir it in.
• Cook until the filling is thick and glossy, then let it cool completely.

Shape the donuts

• Roll the risen dough out on a lightly floured surface.
• Cut out rounds with a cookie cutter and place them on a lined tray.
• Cover and let them proof until puffy and doubled.

Fry the donuts

• Heat the oil in a deep pot to 170 °C.
• Fry the donuts in batches until golden on both sides.
• Drain them on a cooling rack.

Fill and coat

• Toss the warm donuts in the sugar mixed with cinnamon.
• Fill a piping bag with the cooled apple filling and poke a hole in the side of each donut.
• Pipe the apple pie filling generously into each donut.