Heat the milk to just below boiling point. Remove from the heat, add the butter and swirl the pan to melt it. Leave to cool for about 5 minutes until you can comfortably dip your little finger into the liquid, then stir in the lightly beaten eggs.
Put the flour, sugar and yeast into a large mixing bowl and stir to combine, then add a pinch of salt. Make a well in the centre of the flour and gradually pour in the milk mixture, stirring, to form a soft, slightly sticky dough.
Tip the dough onto a lightly floured work surface and knead for 10 minutes, or until you have a soft, shiny dough. Place the dough into a lightly greased bowl, cover with a clean cloth and leave to stand in a warm spot until doubled in size – about 1 hour.
Tip the dough out onto a floured work surface and knead briefly to get rid of any air bubble. Flatten the dough to a 2cm (1in) thickness. Use a 7 cm (3in) round cutter to stamp out the dough or cut into 12 equal parts with a knife, then shape into neat balls. Transfer to two baking trays lined with baking paper, leaving a 5cm (2in) space between each ball. Loosely cover with a cloth, leave to stand for about 30 minutes, or until doubled in size.
Preheat the oven to 180C/ 350F/gas 4. Bake the doughnuts for about 12 – 15 minutes, or until the doughnuts are golden and sound hollow when you tap their underside. Transfer to a wire rack to cool.
Once cool enough to handle, brush the doughnuts with butter then roll in the caster sugar. Put back on the wire rack and leave to cool for another couple of minutes.
While the doughnuts are still a little bit warm, split them almost in half through the centre, tease the 2 halves apart and fill with a scoop of your favourite ice-cream. Sandwich together and serve immediately with plenty of napkins.