1. 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.
2. 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.
3. 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.
Make Ahead: The grilled Portobello mushrooms can stand at room temperature for up to 4 hours.
Notes: If you don't own a cheese slicer, use a swivel-bladed vegetable peeler to make the cheese shavings.