Drizzle extra-virgin olive oil over 4 mushrooms and season with rosemary and salt and pepper. Grill 5 minutes on each side then remove from grill.
While the portobellos cook, cut the remaining portobello into 4 thick slices then thinly slice each piece across. Mix the shaved fresh mushrooms with bread, tomatoes, celery, onion and parsley and dress the salad with the juice of 1 lemon and about 1/4 cup extra-virgin olive oil. Toss and season the salad with salt and pepper, to taste.
Top grilled mushrooms with mounds of the salad and shavings of Parmigiano-Reggiano.