This can be served with game chips and seasonal vegetables if serving as a main course.
The night before, put the walnuts in a small bowl and cover with the wine (the walnuts should be completely covered) . Cover with a lid or cling film and set aside to soak overnight.
To prepare the pigeons, place one on a chopping board with the breast uppermost and the head pointing away from you. Feel for the end of breastbone closest to you, and snip a small hole in the skin with a pair of scissors. Use this hole to help you peel the skin away from the breast meat. Once the breasts are exposed, remove the breasts from the ribs with a sharp knife, following the natural shape of the central breastbone and ribcage with your knife to help prevent leaving any meat behind.
Wash the meat under cold water and pat dry with kitchen paper.
Heat a large frying pan over a medium high heat. Add the olive oil and thyme. Add the pigeon breasts and cook for 30 seconds on each side until lightly coloured. Remove from the pan and cut into thick slices, reserving the cooking juices.
Tip the walnuts and wine into the frying pan, bring to boil and continue to cook until the volume of the liquid has reduced by a third. Season to taste with salt and freshly ground black pepper. Add the sliced pigeon and any collected cooking juices to the pan. Turn off the heat, and cover the pan with a lid or foil and leave to rest for 10 minutes.
Serve the slices of pigeon with the walnut and wine sauce poured over the top.