Trim the top stalks from the fennel, reserving some of the fronds. Trim the root end and peel away any tough outer layers. Cut the fennel bulbs vertically in two. Slice out and discard solid core.

Slice the trimmed bulb very thinly (⅛ inch) on a mandolin slicer, or with a very sharp knife. Place fennel in a colander in sink or large bowl and sprinkle with salt; toss to mix. Let rest for 30 minutes to an hour.

In the meantime, rinse the potatoes to remove any remaining soil. Place potatoes in a large saucepan and cover with cold water and add 1 tablespoon salt. Bring to a low boil until potatoes are tender, but not overdone, about 20 minutes. They are ready when they can be easily pierced with the tip of a knife. Remove the potatoes with a slotted spoon and place them on a cutting board to cool to room temperature. Peel them while still warm. Cut them into large pieces.

Top and tail the beans, rinse them and cook in the same water for 2 to 3 minutes or until tender, but still crisp. Immediately transfer the beans to an ice bath to stop the cooking.

Set the potatoes and beans aside on the counter for at least half an hour, allowing enough time to allow the dressing ingredients to bind, then serve it warm or cold, as a side dish or main course. Garnish with fennel fronds and shaved parmesan if desired.