In a large skillet, heat oil and saute chicken strips over high heat until no longer pink and juices run clear. Remove chicken from skillet and place on a plate.
Peel onion layers apart. Add onions to skillet and saute over low heat for 15 minutes, until they are translucent. Add the reserved chicken strips, balsamic vinegar, red wine vinegar and oregano and cook for 5 more minutes.
Add the flour and stir into the mixture to thicken. Heat through 2 minutes longer, remove from heat and serve.