Preheat grill or grill pan.
Grill portobellos until slightly softened, 3 minutes per side, and place gill side up on a clean work surface.
In a small bowl, mix 4 tablespoons extra virgin olive oil, anchovy paste, vinegar and dry thyme. Divide this mixture onto the gill sides of the portobellos and let stand 30 minutes.
In a mixing bowl, toss clean arugula, remaining 4 tablespoons extra virgin olive oil, lemon juice and sea salt to taste until evenly coated.
Divide the arugula onto 4 plates and top each with a mushroom. Shave the Parmigiano over with a peeler and serve immediately.