Trim the outer leaves off of the artichokes and pare them down to their hearts.
Scoop out the chokes with a spoon, then drop them into water containing the juice of one lemon.
Cut off the tops of the fennel at the base of their stalks. Remove the outer layer of the bulbs. Slice the bulbs paper thin.
Remove the artichokes from the water and slice them very thin.
The salad should be assembled in layers, either on a large platter or on individual salad plates.
First make a layer of the sliced fennel, squeezing lemon juice evenly over the top and then drizzling with about a third of the olive and truffle oils.
Next, make a layer of the artichoke hearts. Squeeze more lemon over the tops of the artichokes, then drizzle evenly with another third of the 2 oils.
Season with salt and pepper.
Cut shavings of the Parmesan and place on top of the artichokes.
Scatter chopped parsley over the cheese, then sprinkle more salt and pepper.
Squeeze more lemon juice over the top, then drizzle with the remainder of the 2 oils.
This salad should be served immediately.