Place split chicken breasts in a large bowl; add lime and water. Let stand while you prepare sauce.
Combine allspice, ginger, pepper and garlic in a mortar and mash to a paste with a pestle or grind in a food processor fitted with the metal blade. Add cayenne, cardamom and cloves and mix again set aside.
Heat a large, deep sauté pan or casserole over medium heat. Without using any fat, add onions and stir constantly until some moisture has been released. Cook, stirring often, until onions are soft and somewhat golden, 30 to 40 minutes. If the skillet gets too dry, add 1 to 2 tablespoons butter. This process is slow, but eventually, onions will take on a golden color.
Add remaining butter and spice paste and stir constantly until butter melts. Keep stirring over medium heat to release flavor of seasonings, 2 to 3 minutes. Stir in 1/2 cup chicken stock and cook for 3 minutes.
Remove chicken pieces from water, shake them dry, and push them into onion mixture. Cover and cook over medium low heat for 7 minutes. Remove cover, stir in another 1/2 cup chicken stock and cook, uncovered, until chicken is tender, about 35 minutes. Add more stock if sauce gets too dry. (It should be creamy-saucy, not thin and runny.).
Sprinkle sauce with salt and more pepper to taste. Carefully stir in eggs, without breaking them up too much or tearing chicken off bone. Carefully spoon chicken, sauce and some egg onto heated plates. Serves 6. Good wine to serve is Cline Cellars Cotes d’Oakly (California: Contre Costa).
Onions, Onions, Onions.