Preheat grill or grill pan to medium-high heat.
Drizzle or brush the eggplant slices with EVOO, season with salt and pepper and grill until tender, 2 minutes each side.
In a food processor, puree the roasted red peppers, balsamic vinegar, EVOO, salt and pepper and pour over the shredded chicken. Add the basil and toss to combine.
Divide the roasted red pepper-dressed chicken among the 8 slices of eggplant. Roll up the eggplant from the narrower end to fat, rounded end. Repeat with all eight slices and set aside.
For the salad, add the chopped hearts of romaine to a large mixing bowl and dress with lemon juice, EVOO, salt and freshly ground black pepper.
Serve the eggplant rolls on top of some of the romaine salad.