Step 1Trim the tough stalks and lacy fronds from the fennel, reserving enough of the fronds to make 1 tablespoon when minced. Trim the ends, then cut the head in quarters lengthwise. Trim out the solid core at the center of each quarter. Using a very sharp knife, cut the fennel quarters in lengthwise slices as thinly as you can. Place them in a work bowl.
Step 2Cut the radishes from the greens and cut them in fairly narrow wedges (one-half inch wide pieces). Add them to the fennel.
Step 3In a bowl, stir together the mayonnaise, the reserved 1 tablespoon chopped fennel fronds, capers, 1 1/2 tablespoons chives and vinegar. Taste and adjust seasoning, adding more vinegar or salt if necessary.
Step 4Add half of the mayonnaise to the salad mixture and toss well to coat lightly. Turn it out onto a platter and arrange the salad in a neat mound.
Step 5Slice the lobster tails in medallions roughly one-half inch thick. Stir gently with remaining mayonnaise to coat lightly. Arrange the lobster medallions on top of the fennel salad. Sprinkle with remaining 1 teaspoon minced chives.