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Light a grill or heat a grill pan. Coat each mushroom with 1 tablespoon of the pure olive oil. Season with salt and pepper. Grill the mushrooms over a medium low fire or in a grill pan until very crusty and lightly charred outside and cooked through, about 10 minutes per side. Transfer the mushrooms to a large rimmed baking sheet, gills up.
Preheat the oven to 400°. In a small bowl, combine the balsamic vinegar and garlic. Brush the mixture over the mushroom gills. Roast the mushrooms for about 5 minutes, or until hot. Slice the mushrooms 1/2 inch thick.
In a small bowl, whisk the extra-virgin olive oil with the lemon juice and thyme. Season with salt and pepper. Toss the pear wedges with 1 tablespoon of the dressing. In a large salad bowl, toss the romaine with the remaining dressing. Add the pear wedges and mushrooms and toss. Top with the cheese shavings and serve.
The grilled Portobello mushrooms can stand at room temperature for up to 4 hours.
If you don't own a cheese slicer, use a swivel-bladed vegetable peeler to make the cheese shavings.