Rinse the pigeons and pat dry with kitchen paper. Cut into four large parts and arrange in a flameproof dish. Finely chop the prosciutto and vegetables and mix well, adding half a glass of oil and the sage leaves.  Add salt and pepper, and allow the pigeons to brown for several minutes, turning frequently. Add a glass of red wine and allow to evaporate.
Then add the pitted, finely-chopped olives, and the capers, well rinsed. Cover and leave to cook over a low heat for about 50 minutes, adding the chicken broth as necessary. Take off the lid, turn up the heat and allow the sauce to thicken for a few minutes. Serve the pigeons in their sauce.