3rd Monitor question using TBD

I have a 2012 Air that I pair with a TBD display at home, but while at work I am using a Mini Display port to DVI adaptor that allows me to run a external monitor at my desk.

After mentioning that my workflow at home is so much better than in the office, my employer has ordered me a TBD for the office and it is currently shipping to me. My question is, can I use the mini displayport to DVI adaptor on the back of the TBD to add an additional monitor? (Air conected to TBD, with 2nd external monitor using adaptor to plug into the back of the TBD). Has anyone tried this and does it work?

 

Thanks for the reply.

Will any peripheral work or is there a specific one that is inexpensive and proven to work? The $199 device you named also? Thanks again