Use FaceTime to record a video with screen sharing?

I need to record a presentation, going back and forth between me talking and demonstrating my company's new iPhone app. The app runs in the Xcode iOS simulator on my MacBook, and my MacBook has a FaceTime camera (720p!). Is there a way to use FaceTime to record a video? Skype has screen sharing. You can just click a button and the caller sees what's on your computer screen. Then you click again to return to the caller seeing you talking. Does FaceTime have a screen sharing feature? Is there another app I should be using?