Make the dressing: Pound half the toasted walnuts with a mortar and pestle until just broken down (you don’t want this to be a paste). Alternatively, finely chop the walnuts. Place the broken-down walnuts and olive oil in a small sauce pot and warm them over low heat for a few minutes. Remove from heat and let cool to room temperature. This will allow more of that walnut flavor to permeate the dressing.
While the oil is resting, combine shallot, vinegar, mustard, and a pinch of salt and pepper in a small bowl. Add the walnut oil and taste the dressing for seasoning.
Build the salad: Combine the chicories and celery in a bowl with some salt and pepper, the rest of the reserved walnuts, and a few spoonfuls of the vinaigrette. Toss the salad, taste, and adjust seasonings accordingly. Plate, and shave curls of Parmesan on top of each serving.