Heat olive oil in a frying pan. Cook broccoli, potato, and onion in oil until tender, about 5 minutes. Add chili powder, garlic, salt, and chicken to the vegetables. Saute until chicken is cooked through, stirring constantly, 10 to 15 minutes. Transfer to a food chopper, chop entire mixture until fine.
Divide mixture between the tortilla wraps. Top each with a slice of American cheese.
Fold ends toward the middle, and roll the wrap away from you.