So I'm thinking of heading across the pond for 7 - 10 days at the end of November (taking advantage of the American Thanksgiving holiday). Any suggestions on where to head for some alpine climbs? Looking for something in the AD - D range.

Would the Sierra Nevada or Pyrenees be in yet at that time? Would the weather in the Alps be stable? What about the Tatras?

What kind of activity do you seek ? Well, it all depends if you want snow or not. By AD and D you mean rock climbing ?
The Spanish Pyrenees are certainly a very good choice. See for example the peak pages submitted by user OJO. No idea for Sierra Nevada but could be.
For the Alps, it all depends on the weather. In the southern Alps if the weather will be warm till then, you may still find nice autumnal conditions.
Maybe you should consider all southern ranges in general. I bet Greece could also be.
For the Tatras, it's already the winter at that time and it can be (or not be) very dangerous. Read the posts about the recent accident of one SPer.
Eric

Maybe you want to consider the Grand Sasso area, too.