With a mandoline slicer or a sharp knife, cut fennel into paper thin slices. With a peeler, peel lemon zest in strips, then slice into thin slivers. Juice the lemon, straining seeds. You should have about 1/4 cup of juice. Toss fennel with 1/4 teaspoon sea salt, half the lemon zest, and all of the juice, and set aside. Allow fennel to marinate for about an hour at room temperature.
To plate, lift fennel out of the liquid (leaving the accumulated juices behind) and arrange in an s-shape on a large platter. Tuck mozzarella into the fennel. Lift cherries out of the balsamic, and scatter wherever there's a space on the platter. Tuck reserved basil leaves throughout. Finally, drizzle generously with basil dressing. Taste, and sprinkle with sea salt as needed. If you like, add pepper.
Leftover salad can be covered and stored in the fridge (though not as pretty, it will taste delicious). Leftover dressing will keep sealed in the fridge for two days, though it will lose some of its vibrant color.