In a small bowl whisk together mayonnaise, yogurt, horseradish, Cognac, onion, ketchup, and lemon juice. Season with salt and pepper, to taste. Cover and chill the sauce.
Plunge the lobsters into a large kettle of boiling salted water and boil them, covered, for 10 minutes. Transfer the lobsters with tongs to a bowl and let them cool until they can be handled. Crack the shells, remove the meat, and cut it into 3/4-inch pieces. Transfer the lobster meat to a large bowl and chill it, covered. The lobster cocktail may be prepared up to this point 1 day in advance.
Halve the mangoes by cutting just to the sides of each pit. Using a 3/4-inch melon-ball cutter, scoop the flesh from the mango halves (There should be about 2 cups). To the lobster meat add the mango, celery, avocado, sliced endives, chives, and sauce and toss mixture until it is combined. Divide the lobster mixture among 12 chilled small glasses and garnish each serving with 1 of the endive leaves and 2 of the whole chives.