I'm an extrovert too, would talk to virtually anybody but not to my T. What we should be talking about- the weather? IDK, it would just not feel natural to me to come up to my T and have a "normal discussion"- without all the me-me-me stuff.
So no way I would choose to interact with him - on the few (2 actually) occasions I've met him outside his office I smiled, said hi and went away.
However, if you wouldn't mind all this, would even like to and your T says it's OK then go for it