For the Doughnuts; Warm the milk for about 15 seconds in the microwave. You don’t want it to be hot, just slightly warmer than room temp. Pour the milk into the bowl of a stand mixer then sprinkle the yeast on top and stir it around. Let it sit for about 5 minutes.
Add the flour, sugar, salt, eggs and egg yolk. Beat with a dough hook on low speed for a couple of minutes until the dough comes together. With the mixer on medium add the butter a spoonful at a time until it is all fully incorporated.
Grease a large bowl with a tiny bit of oil or oil spray. Place the doughnut dough into the bowl. Cover tightly with plastic wrap and refrigerate overnight.
For the custard: Whisk together the egg yolks, flour and sugar until very thick and creamy and light in colour.
In a saucepan on low-medium heat, heat together the milk and vanilla until it reaches a simmer. Take it off the heat and while whisking continuously, slowly dribble the hot milk into the eggs. You must do this slowly so the eggs heat up slowly and don’t scramble.
Return the saucepan to a low heat. Stir constantly until you have a very thick paste. Make sure to keep stirring so that you don’t get any lumps.
Push the custard mixture through a sieve into a bowl to remove any lumps, then refrigerate until completely cool.
Whip the cream, then fold it into the cold custard. Now it’s ready to pipe into your doughnuts. Refrigerate until required.
To cook the Dougnuts; Line a baking sheet with baking paper. On a lightly floured surface, gently roll out the dough to about 12mm thick (1/2 inch) and make it about 30cm square (12 inch square). Use large round cookie cutters to cut out 9 rounds from the dough and lay them on your baking paper.
Put them aside in a warm place for 1-2 hours until they are doubled in size and puffy looking.
Lay out a couple of layers of paper towel on a plate or tray. Attach a thermometer to the side of a deep pan and pour in the oil. Heat the oil to 180C. Gently place the doughnuts one at a time into the hot oil, in batches of 3 only. Deep fry for 1-2 minutes, then flip over and fry the other side for 1-2 minutes. Repeat with the other 2 batches of 3 doughnuts at a time making sure the oil is at the right temperature before starting. Remove them from the oil using a slotted scoop allowing the oil to drip off and transfer them to the paper towel until slightly cooled, then transfer to a cooling rack.
Once they are cool, cut a small slit in the side and place the tip of your piping back into the slit and gently squeeze in your filling until you feel the piping bag being pushed back. Gently pull out the piping nozzle but filling the doughnut right to the outside.
For the Caramel Icing: Add the butter, sugar and milk to a small saucepan, stirring over medium heat. Once the butter has melted and the sugar dissolved, bring to the boil, then reduce the heat and simmer for 1 minute. Remove the pan from the heat and cool for 10 minutes. Add the icing sugar and whisk in until fully incorporated.
Place your doughnuts on a cooling rack, set over the top of a baking tray to catch any drips. Pour the icing over the top of the doughnuts while it is still a bit warm.