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Place flour, sugar, yeast and salt in the bowl of an electric mixer and stir to combine.
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Add milk, egg and egg yolks. Mix with a dough hook until a sticky dough forms, and mix for another 3-4 minutes. If doing this by hand, tip the dough onto a lightly floured bench and knead for 10 minutes.
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Slowly add the butter, one piece at a time, over the course of 8-10 minutes until the butter is completely combined and the dough is silky smooth.
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Place dough in a bowl and cover with a damp tea towel. Leave to rise in a warm place until almost doubled in size, about 40min to an hour.
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Roll the dough out on a lightly flour bench to 3cm thick. Cut out rounds using a cookie cutter (I managed to cut out 14 donuts, and my cutter was...)
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Place on a baking tray lined with baking paper and leave to prove for another 10-20 minutes. The dough should feel light and fluffy when pressed, and a finger indentation should spring back slowly (if a finger print stays indented, it is over proved).
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At the same time, heat the oil in a large pot to 165-75°C on a candy thermometer. Cook the donuts for 1-2 minutes on each side until golden brown. Remove with a slotted spoon onto a wire rack to drain.
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Allow to cool, then poke a hole into each donut with your finger to create a cavity/space inside.
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Fill a piping bag with the pastry cream you wish to fill them with, and gently fill the donut until it expands against your hand and the filling oozes out of the hole.
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Spoon over the chocolate glaze or caramel sauce (heat gently in a microwave if it isn’t runny enough), and top with chopped hazelnuts and a dusting of icing sugar.