Whisk together the mayonnaise, the yogurt, the cognac, the ketchup, the lemon juice and add salt and pepper to taste.
Chill the sauce, covered.
Plunge the lobsters into a large kettle of boiling salted water and boil them, covered, for ten minutes.
Transfer the lobsters to a bowl and let them cool until they can be handled.
Crack the shells, remove the meat, and cut it into three centimetre pieces.
Transfer the lobster meat to a large bowl and chill it, covered. (The lobster cocktail may be prepared up to this point one day in advance.)
Halve the mangoes by cutting just to the sides of each pit and, using a three centimetre melon-ball cutter, scoop the flesh from the mango halves. (There should be about two cups.)
To the lobster meat, add the mango balls, the celery, the whole chicory, trimmed and sliced thin crosswise, the chopped spring onions and the sauce and toss the mixture until it is combined.
Divide the lobster mixture among 12 chilled small glasses and garnish each serving with one of the chicory leaves and two of the whole spring onions.
Important ? Serve well chilled.