Spotify's connect future is really convenient of you have Spotify enabled devices around the house. Especially a Chromecast on your TV right? But how can we really play our favorite song without a smartphone around?
Cast supported devices
As mentioned here the only supported devices you Cast from are: Android, iOS, Chrome. Even though Spotify desktop uses Chrome Framework currently doesn't really support Cast SDK.
Solution
Even though we can't (yet) really play a song through Spotify Desktop, we CAN from Desktop. You see Spotify has the web player which can really be our workaround! You just login in your account through you Desktop's Chrome, and you if you click on the Devices...Voila!
When you start playing to your Chromecast device you can open up you Desktop app and control your device!