Preheat the oven to 200C/400F/Gas 6.
For the scones, mix together the flour, butter, egg and just enough milk to bind in a food processor to form a smooth, firm dough.
Roll out some scone dough onto a flat surface. Mix in the cheese (reserving a little for later) and basil and roll out the dough to 2½cm/1in thickness.
Using a medium-sized chefs' ring or cutter, cut out rounds and place onto a baking sheet. Brush with milk and top with the reserved cheese.
Place into the preheated oven for 15 minutes, or until risen and golden.
Meanwhile, for the onions, heat the oil in a saucepan, add the onions and fry them until soft.
Add the sugar and let the onions caramelise for 2-3 minutes.
Add the balsamic vinegar and reduce till thick and syrupy.
Serve the warm scones with balsamic onions.