Prepare a hot grill fire or heat a grill pan over medium until hot. With a mallet or cast-iron skillet, pound chicken breasts to ½” thickness. Rub with oil, salt and pepper. Place on grill or in pan and cook, flipping once, until browned on both sides and cooked through, about 5 minutes per side. Internal temperature should be at least 165°F. Let cool slightly and thinly slice.
In a large, dry skillet over medium, heat tortillas one at a time, flipping once, until heated through and pliable. Transfer to a plate and warm remaining tortillas, stacking them on top of each other.
Serve chicken on tortillas with lettuce and tomatoes. Drizzle with dressing, and sprinkle with Parmesan.