"An" is correct because it's the first time you're mentioning him to us. If you continue your story, you'll say "the" Indian Chef, because you've just specified him. "I ate dinner out at a local Indian restaurant run by an Indian chef and his Japanese wife." (continuing:) The Indian chef (that I just mentioned) came out of the kitchen and said "hi" to the diners.

"I ate dinner out at a local Indian restaurant run by an Indian chef and his Japanese wife." Yes, this is correct. If you wrote "the Indian chef", that would mean I (the reader) should already know which chef you mean. Here's an example of "the": "I ate dinner out at a local Indian restaurant. The chef-owner is Indian and his wife is Japanese." Both you and I know "which" chef-owner, as it's obviously the chef-owner of the restaurant you just mentioned. By the way, I used "is", assuming that this happened very recently and that you might be recommending this restaurant to me.

Using the indefinite article ("an Indian chef") is correct. You would use "the Indian chef" if you wanted to be particular (ie. definite) in your specification of the Indian chef. Eg. There are three chefs: an Indian, an Englishman and a Brazilian. The Indian chef owns the Indian restaurant. Hope that helps!

an

An is Correct.