Chop the onion and two of the figs coarsely and put them in a saucepan large enough to hold all the fennel. Add the chicken stock and salt. Bring the mixture to a simmer.
Meanwhile, cut the dark green ends away from the fennel and reserve for other use or discard. Cut the rest lengthwise into quarters and add them to the pan.
Cover the pan tightly and let the fennel cook at a very low simmer until it is tender, about 30 minutes. (Note: It is not necessary to cover the fennel with liquid; it will cook evenly in the tightly covered pan.)
Remove the fennel from the pan juices.
Puree the pan juices in a food processor or blender, adding lemon juice to taste. Taste for seasoning.
To serve, cut the tips from the remaining figs and cut them lengthwise into thin slices, leaving them attached at the base. They will fall open into a fan shape. Place one at the edge of each of eight small salad plates.
Overlap two fennel quarters at the center and spoon some of the puree over the fennel.
Garnish the rest of the plate with a handful of greens and shaved Parmigiano-Reggiano.