have a 2nd monitor and move the steam client over to it.
also, you'll have to press shift + tab and click settings on the bottom to setup your steam broadcasting settings (you probably can't watch your own stream unless you set it to anyone, because you aren't friends with yourself)
If you alt tab the stream will close, but if you run in windowed mode it will stay open.
After you start broadcasting, if you shift tab there will be a new Watch button that will let you see your own broadcast (it just opens a steam web browser) but only while you're shift tabbed.