An hour ahead, soak the cockles in fresh water to purge any sand. Rinse the other shellfish thoroughly in several changes of water.
Place the mussels in a large pot over high heat until they open. Transfer to a bowl, then repeat with the clams and cockles, cooking each batch separately.
Wash, dry, and finely chop the herbs—you'll need about 1 tablespoon of each.
Shell the cooked shellfish and place in a large salad bowl. Add the chopped herbs and gently mix.
In a small bowl, combine the olive oil with the grated lime zest. Squeeze in 2 tablespoons of lime juice, season with salt, and whisk into an emulsion.
Blanch the tomato in boiling water for 1 minute, peel, and dice the flesh.
Peel and thinly slice the onions into strips. Add the diced tomato and onions to the shellfish. Pour over the dressing and toss well. Adjust seasoning to taste.
Wash and spin-dry the lettuce leaves. Arrange on serving plates, mound the shellfish salad in the center, dust lightly with powdered ginger, and serve immediately.
Blanc de Chine is a traditional name for finely powdered ginger.
