Peel the carrot, onions, garlic and shallots and chop.
Remove the wings, but not the legs. Cut the pigeon in half lengthwise and season each half with salt and pepper.
In 1 tablespoon of oil, brown the pigeon halves. Add to the pan ¾ of the prepared vegetables and cook. Add the bouquet garni, pink peppercorns and 1 liter (approximately 1 quart) of water to the pan. Over high heat bring to a boil and cook for about 1 hour, skimming from time to time. Pour the liquid through a fine sieve.
In a pot, combine the honey and remaining vegetables and stir until caramelized. Deglaze with vinegar and boil for 5 minutes until syrupy and add to the pigeon stock. Let simmer for 20-30 minutes and pour through a fine sieve. Season with salt and pepper.
Heat the remaining oil and butter in a pan. Brown the pigeon halves on all sides. Place on a baking sheet and place in a preheated oven at 250°C (approximately 475°F) and cook for 6-8 minutes. Place some sauce on a plate and arrange the pigeon halves on top. Sprinkle with cumin and serve.