Heat a flat grill plate until very hot. Trim 1 cm (½ inch) off the top of each onion and grill until burnt. Slice the burnt 5 mm (¼ inch) off and place the burnt pieces in a saucepan. Return the onions to the grill and burn again. Repeat this until all of the onion is burnt.
Add the water and stock to the pan and simmer very gently for 2 hours or until it begins to taste bitter with a touch of sweetness and has reduced to about 250 ml (9 fl oz/1 cup). Pass through a fine sieve and then again through muslin (cheesecloth).
Combine the vinegar and sugar in a small saucepan and heat until the sugar has dissolved.
Add the bitter onion stock to the pigeon sauce, a little at a time, until the correct balance is achieved — it should taste rich and intensely savoury from the pigeon stock with a fair amount of bitterness from the onion. Bring to the boil and add a few drops of the sherry vinegar syrup until there is a hint of acid.