To make the doughnuts, warm the milk in a saucepan. Add the yeast and stir until dissolved.
Pour into the bowl of a stand mixer fitted with a paddle beater. Slowly beat in the flour and cocoa until combined. Gradually add the butter and then the egg.
Allow the dough to prove until doubled in size.
Knock back the dough and divide into 12 equal pieces and shape into balls, place on a lined baking tray and leave for 1 hour. Meanwhile, line another baking tray with baking paper and scatter over the caster sugar.
Preheat a deep-fat fryer to 180 C. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Deep-fry the doughnuts for 3–4 minutes, until golden-brown. Remove using a slotted spoon and carefully roll in the caster sugar.
To make the coffee sauce, warm the milk in a medium saucepan with the ground coffee, espresso and coffee liqueur.
Put the yolks, cornflour and sugar in a bowl. Pour over the warm coffee mixture and whisk.
Return the mixture to the pan and whisk, over the heat, until the sauce has thickened. Remove from the heat and stir in the cream.
Put the doughnuts in a serving bowl, pour over the coffee sauce and serve.