- In a small bowl whisk together mayonnaise, yogurt, cognac, ketchup, lemon juice, and salt and pepper then chill covered.
- 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 ¾-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 mangos by cutting just to the sides of each pit and, using a ¾-inch melon-ball cutter, scoop the flesh from the mango halves (about 2 cups).
- To the lobster meat add the mango balls, celery, whole endives, trimmed and sliced thin crosswise, the minced chives and sauce, then toss mixture until 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.