I have found a lot of versions. Which do you think is most appropriate and why?
甕缸雞/桶仔雞
Crock chicken
Barrel chicken
Urn chicken
Clay pot chicken
Claypot chicken
Claypot roasted chicken
Clay oven chicken
Jar-roasted chicken
Wongzi chicken
Wonggang chicken
Kiln-roasted chicken
Earthenware chicken
Clay pot chicken seems to get most hits on Google search, but the pots or not necessarily made of clay.
Sounds most normal.
But are those monstrosities in the image pots? Look more like Chinese wine jars or urns?
Pot’s a pretty generic term. I don’t anybody’s gonna say, “No, but that’s an urn.”
I would say in English “Taiwanese roasted chicken” or “BBQ chicken” or if more details are desired “Would you like try local roasted/BBQ chicken that was roasted in a barrel”
An English speaker is going to think of something completely different if you say this. It’s almost as bad as calling 蔥油餅 “Chinese pizza.”
I think it’s important to have the “pot,” or whatever you want to name it, in the term.
Or Taiwanese hamburger for 刈包?
If you call it this, people will think it tastes good though. 
There’s a place in Taipei (one of like, two in the entire Taipei area) that has an English sign calling it ‘bucket chicken’. I think it fits, it reminds me of KFC, and it avoids ideas like Tandoori. Usually though I wouldn’t bother with the translation and just call it thang-a-ke, since no English approximation will get the full meaning across of what type of food, what kind of restaurant, cooking style, culture, etc.
甕 is translated to Earthen Jar, Jar, or Urn.
Typically, urn is not for cooking.
So Jar Chicken or Earthen Jar Chicken.
I think because Chinese Hamburger was already taken.
