Summary.: The Braess Paradox is the counterintuitive fact that creation of a shortcut may make travel slower. As each driver seeks to minimize his/her travel time, the shortcut may become so popular that it causes congestion elsewhere in the road network, thereby increasing the travel time for everyone. We extend the paradox by considering a shortcut that is a single-lane but two-way street. The conflict about which drivers get to use the single-lane shortcut is an example of a game theoretic situation known as Chicken, which merges with the Braess Paradox into the novel Chicken Braess Paradox: meeting traffic may make travel quicker.