To make the stock, start by chopping the reserved pigeon carcasses into three with a large, heavy knife. Heat the oil in a large saucepan over a medium heat. Throw in the carcasses and brown them well, then pour in enough water to cover the bones (about 1.25 litres). Add the peppercorns, skin and trimmings from the onion and the stripped stalks from the thyme you will use in the pie filling. Bring to a gentle simmer and cook for an hour with the lid off (after this time you would be hard pushed to extract any more flavour from the bones). Strain the stock into a bowl, then return it to the saucepan and reduce it to 500ml. Season it with the Worcestershire sauce and salt, if needed, then leave it to cool. The stock should be delicious in its own right.
While the stock is doing its thing, make the pastry. Mix the flour and salt in a large bowl and stir in the butter with a round-bladed knife. Slowly add enough of the water to bring the mixture together into a soft dough. Turn out on to a floured work surface and knead very lightly. Form into a flattish ball and wrap in baking paper. Chill in the fridge for 30 minutes.
Meanwhile, make the pie filling. Take a large frying pan and drop in 25g of the butter. When it has melted, add the bacon, carrots and onion with the bay leaf, cloves and thyme leaves. Cook all together over a medium–low heat, stirring occasionally, until the carrots and onion have become tender (about 15 minutes). Add the vinegar and cook until it has evaporated. Put the contents of the pan in a bowl to one side. Wipe out the pan with kitchen paper.
Preheat the oven to 180°C fan/200°C/Gas 6. Pull the skin from the pigeon breasts and discard, then chop the breasts into large chunks, about three pieces per breast. (The size is important as, chopped smaller, they would overcook by the time the pastry is done and will be tasty but tough.) Sift the flour into a mixing bowl and add some salt, combining everything together briefly. Toss the pigeon pieces in the seasoned flour until well covered.
Melt the remaining butter in a large frying pan and when it is smoking fry the pigeon bits hard, for no more than 2 minutes, tossing regularly so as to brown the pieces evenly on all sides. Again they need to colour fast so as not to spend too much time in the pan, so really make sure the pan and butter are truly hot. Combine the pigeon pieces with the carrot, bacon and onion. Place a pie bird in the middle of a 1.25-litre pie dish. Add the filling; it should come slightly above the level of the rim to prevent the pastry from sagging. Pour over the cooled stock so that it comes up to just below the rim of the dish.
Take the pastry from the fridge about 15 minutes before using it. Roll it out on a floured work surface to slightly thicker than a £1 coin and 5cm larger than the pie dish. Brush the rim of the dish with beaten egg. Cut two or three 2cm-wide strips from around the edge of the pastry and press these on to the rim all the way around. Brush with more egg and carefully lift the remaining pastry over a rolling pin and on to the pie dish, making a small slit with the point of a knife to allow the pie funnel to pass through the pastry. Press the edges to seal, then trim neatly. Crimp the edge, if you like, to give the pie a decorative finish. Glaze with more egg. Bake the pie on a baking tray until its roof is dark golden brown and the filling is hot (about 30–35 minutes).