Preheat barbecue grill to medium-high. (Alternatively, heat cast iron grill pan over medium-high heat and brush with oil.) Cut romaine hearts in half, lengthwise. Brush cut sides with olive oil. Place romaine hearts, cut side down, on grill and cook, turning once, until charred and slightly wilted, about 3 minutes, depending on heat. Transfer, cut side up, to serving platter. Scatter roasted vegetable salad overtop, dividing evenly among romaine hearts. Sprinkle with grated pecorino and drizzle with extra-virgin olive oil, balsamic vinegar, and chopped fresh basil.
