Mix olive oil, lemon juice, zest, garlic, salt, and pepper in a shallow dish. Add chicken and marinate 2-6 hours if time permits.
Make dressing: Combine olive oil, mayo, lemon juice, Dijon, Worcestershire, garlic, anchovy, salt, pepper, and half the Parmesan.
Grill chicken 5-6 minutes per side until cooked through. Let cool slightly, then cut into bite-sized pieces.
Toss chicken, romaine, and onion with dressing. Add remaining Parmesan.
Fill each wrap with 1½ cups of mixture, tuck ends, and roll closed.