Drain the crab meat well in a sieve and pat it lightly dry with paper towels. Excess moisture will later dilute the mayonnaise into a sauce that runs off the lettuce instead of clinging to it.
Toast the flaked almonds in a dry frying pan until lightly golden. Keep a close eye on them and shake the pan regularly, as thin flakes can go from golden to burnt in half a minute. Let them cool on a cold plate.
Cut the fennel bulb in half, remove the tough core, and shave the halves paper-thin. Cut the unpeeled apple into thin matchsticks and toss them immediately with some lime juice to prevent them from turning brown.
Whisk the mayonnaise with the finely grated zest and juice of half a lime, salt, and black pepper into a thick, creamy dressing. If it is too stiff to spoon, whisk in a teaspoon of cold water.
Gently fold the crab meat into half of the lime mayonnaise and stir in finely chopped dill. Stir carefully, as crab meat easily breaks down into mush if handled too roughly, losing the fibrous texture that makes it special.
Arrange the little gem leaves loosely on a plate with the fennel and apple in between. Spoon the crab salad on top and add slices of avocado. Drizzle the remaining mayonnaise over it and finish with the flaked almonds, extra dill, and black pepper.