Preheat the oven to 400 degrees. Lightly brush a rimmed baking sheet with olive oil, and arrange the prosciutto in a single layer. Place in the oven, and bake until crisp, 5 to 10 minutes. Remove from the oven, and let cool on a wire rack.
Combine the olive oil, balsamic vinegar, and thyme in a bowl; season with salt and pepper, and whisk to combine. Add the romaine and shaved Pecorino Romano; toss to combine. Arrange on 4 plates; serve topped with the prosciutto crisps.