Pre-heat the oven to gas mark 4, 350°F (180°C). Joint the birds into two breast joints and two leg joints each and stew the rest of the carcasses in a little water to make stock for the gravy. Cut the beef into small pieces and line a deep 8 in. (20 cm) pie-dish with these. Lay the pigeon joints on top, cover with water, add salt and pepper, then cover the pie-dish with greased paper or aluminium foil. Place in the oven and simmer for 1½ hours.
Remove from the oven and raise oven temperature to gas mark 6, 400°F (200°C). Cover the pie with the shortcrust pastry, brush the top with beaten egg, put back into the oven and bake until the pastry is golden brown.
Make a gravy by mixing 1 dessertspoon of cornflour with a little cold water and add to 10 fl oz (300 ml) of the warmed stock. Allow to thicken while stirring, season and serve with the pie.