Prepare the salad greens. Pull away the two or three outer layers of the radicchio than halve it lengthwise dividing the core. Carefully cut away the core but keep the radicchio in tact. Slice the radicchio lengthwise about 1/4 inch wide. Place in a bowl and immerse in fresh, cold water. Hold twenty minutes. In the meantime, wash the arugula, dry well and slice the endive crosswise 1/2-inch wide. Transfer to a large salad bowl. Drain the radicchio and immerse again in fresh cold water. Hold another twenty minutes then drain and dry in a salad spinner (by doing this, the water extracts a lot of the bitterness of the radicchio). Toss with the arugula and endive. Put the apricots in a small bowl. In a small pot add the honey and the sherry vinegar and warm to combine then whisk in the olive oil (have more olive oil on hand to adjust the balance of the vinaigrette). Pour over the apricots and adjust the seasoning with salt and pepper.

Pour the vinaigrette over the salad greens and toss to coat the greens. Portion the salad on four chilled salad plates or bowls then top each with any apricot left in the bowl. Divide the parmesan cheese over the top of each salad then finish with the walnuts. Serve immediately.