Preheat grill or grill pan. Drizzle chicken with EVOO and season with salt and pepper.
Grill chicken breasts, about 5 to 6 minutes on each side, then thinly slice.
Coat sliced chicken in salsa verde.
Place tortillas on the grill and char on each side. Stack the tortillas and wrap in foil to keep warm.
Place chicken in the tortillas, top with avocado and cheese, then wrap and roll into burritos.