Preheat your grill to medium-high. Brush the chicken with the oil; season with salt and pepper. Grill 4 minutes per side, or until cooked through.
When cooked, top each breast with half a roasted red pepper, and 1/4 cup of cheese. Cover the chicken and cook until the cheese is melted. Spread the mayonnaise and pesto on the buns. Top bottom buns with the salad greens. Set on the chicken and top buns and serve.