Heat together the water, butter and milk until the butter has melted.
In a separate bowl, mix together the flour, caster sugar, salt and yeast.
Pour the warm butter mixture into the flour and add the egg.
Stir to combine and knead for 5-8 minutes until the dough is smooth and soft. If the dough is sticky, add a little more flour.
Place in a clean bowl and cover with a damp tea towel. Allow to rise for 1 hour.
To make the filling, heat the milk and vanilla pod until it just starts to boil.
In a separate bowl, whisk together the sugar and egg yolks until light. Add the flour and salt and mix well.
When the milk has reached boiling point, pour it slowly into the egg yolk mixture whilst whisking.
Return the mixture to the saucepan and allow to cook over low heat whilst continuously stirring until thick and glossy (around 5-10 minutes).
If there are any lumps, strain the filling through a fine sieve then place in a clean bowl. Push cling wrap onto the surface of the custard/filling to prevent a skin from forming and place in the fridge. Allow to cool completely.
When the dough has risen, knock it down. Place the dough on a floured surface and roll out to approximately 2cm thick. Using a round cookie cutter or small glass, cut circles out of the dough then place on a tray lined with baking paper. Allow to rise for another 30 minutes.
Heat the oil then fry the doughnuts in batches until golden brown and cooked through.
Remove from the oil and drain on kitchen paper.
Allow to cool to room temperature. Place the chilled filling into a piping bag fitted with a plain nozzle then pipe the mixture into the doughnuts. Alternatively you could make a slit in each doughnut and fill with a spoon but this is a little messier than working with a piping bag.
Once all the doughnuts have been filled, place the sugar for the caramel in a frying pan over medium heat.
Allow the sugar to start caramelising, stirring gently until all the sugar has dissolved and the sugar is amber in colour.
Remove from the heat and carefully place the doughnuts into the warm caramel and gently shake the pan to coat them. Carefully remove the doughnuts from the caramel with a pair of tongs and place them back onto the baking paper. Allow to set then serve.