Grate 1/2 teaspoon of orange zest, then cut the rind and pith away from the orange. Segment the orange, over a bowl, making sure to cut on each side of the membrane. Squeeze the remaining membrane and reserve the juice.
Whisk together the lime juice, mustard, 4 teaspoons reserved orange juice, 1/2 cup extra-virgin olive oil and 1/2 teaspoon orange zest. Season with salt and pepper, to taste.
Heat a grill to medium.
Brush the cut flat sides of the fennel bulbs with olive oil and season with salt. Put on the grill and cook until you see grill marks. Remove them from the grill to a cutting board and let cool. Remove and discard the core. Thinly slice the fennel bulb vertically. Add the fennel slices to a glass bowl along with the shaved celery and the whole parsley leaves. Add the vinaigrette, toss to combine and season, to taste, with salt and pepper. Divide the salad among 4 individual serving plates. Using a vegetable peeler, shave 4 strips of Parmigiano-Reggiano and put 1 strip on top of each salad. Arrange the orange segments on top of composed salad and serve.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.