Preheat an outdoor grill for high heat, and lightly oil the grate.

Brush each romaine lettuce half with 1/2 teaspoon olive oil.

Cook romaine hearts, cut-side down, on the preheated grill until the cut-side is slightly caramelized but the inside is still cool and crisp, about 2 minutes. Transfer lettuce, cut-side up, to a platter. Sprinkle Parmigiano-Reggiano cheese over lettuce and drizzle balsamic vinegar, sherry vinegar, and 2 tablespoons olive oil over the top.