Preheat the oven to 210C/425F/Gas 7.
Season the cavities of the pigeons with salt and freshly ground black pepper and place a sprig of thyme and a half a garlic clove inside each one.
Heat an ovenproof frying pan until smoking, add the butter and the oil and heat until the butter is foaming. Add the pigeons and fry on both sides for 1-2 minutes, or until just golden-brown all over. Place the pan into the oven and roast for 7-8 minutes, or until the pigeons are a deep golden-brown and cooked through. Remove the pigeons from the oven, cover with aluminium foil and leave to rest for ten minutes.
For the baby leeks, bring a pan of salted water to the boil, add the leeks and cook for 2-3 minutes, or until just tender. Drain and keep warm.
For the mushroom fricassée, melt two-thirds of the butter in a frying pan until foaming, then add the shallots and cook for 2-3 minutes, or until just softened. Add the wild mushrooms and sweat for another minute, or until softened.
Add the Madeira to the pan and simmer for 4-5 minutes, or until the liquid has reduced by half. Add the chicken stock and reduce again by half.
Stir in the cream and whisk in the remaining butter until smooth and creamy, then season, to taste, with salt and freshly ground black pepper. Stir in the chopped chervil and keep warm.
For the rösti galettes, grate the potatoes into a clean tea towel and then squeeze out all of the excess moisture.
Heat a frying pan until smoking, add the oil and shallots and cook for one minute, or until softened but not coloured. Add the bacon lardons and fry for another 1-2 minutes, or until the bacon is golden-brown and crisp.
Add the grated potatoes to the pan and mix well. Stir in the clarified butter, reserving 1-2 tablespoons, then season with salt and freshly ground black pepper.
Heat a large frying pan until hot, add the reserved clarified butter and place four metal 8-10cm/3-4in chefs' rings into the pan. Press the potato mixture into the rings and fry for 2-3 minutes, then, using a spatula, carefully turn the chefs' ring over and fry the rösti on the other side for a further 2-3 minutes (if needed, press the rösti down with the spatula so that it touches the pan), or until golden-brown on both sides and tender all the way through. Remove the potato galettes from the rings and place onto a tray with a clean tea towel underneath to drain off excess fat.
To serve, carve the breast and legs from each pigeon. Place a rösti galettes into the centre of each of four warmed serving plates, then arrange the pigeon breast and legs on top. Place the baby leeks next to the galette, then spoon the wild mushroom fricassée over and around.