Preheat grill or broiler. Drizzle the portobello mushrooms with extra-virgin olive oil. Place the mushrooms on the grill so that they are centered near the heat. Grill until soft and nicely cooked, and barely charred, about 3 to 4 minutes per side. Sprinkle some lemon juice on the mushrooms after they have been charred and splash them with red wine vinegar.
In a mixing bowl, toss the pea sprouts, olive oil, and remaining lemon juice, so that the sprouts are evenly coated in the liquid. Season, to taste, with salt and pepper, and divide among 4 plates. For a nice effect, stack the sprouts like little hay stacks.
Using a peeler, shave cheese over and around salad. Using the mandolin, slice the domestic mushrooms in paper thin slices over and around the salad. Place the portobellos on each haystack and serve.