How to Make It
Bring a large pot of salted water to a boil. Add the lobsters and cook until they are bright red, about 7 minutes. Reserve 2 tablespoons of the lobster cooking water, then drain the lobsters. Transfer the lobsters to a large rimmed baking sheet and let them cool slightly.
Twist the lobster tails from the bodies. With scissors, cut down the length of each tail shell and remove the meat. Cut the tails in half lengthwise and discard the dark intestine. Crack the claws and remove the meat. Cover the lobster and refrigerate.
Meanwhile, in a large bowl, mix the cold water with 1 tablespoon of the lemon juice. Add the fennel and refrigerate for about 25 minutes, until the fennel is very crisp.
In a medium bowl, mix the mayonnaise with the tarragon, the reserved lobster cooking liquid and the remaining 2 tablespoons of lemon juice. Season the dressing with salt and pepper.
Drain the fennel and pat dry with paper towels; transfer to a large bowl. Add the arugula, shallot and cilantro leaves and toss well. Add all but 1/4 cup of the dressing to the bowl and toss well.
Arrange the salad on plates and top each one with a claw and half a lobster tail. Drizzle the lobster with the remaining dressing and serve at once.