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But I would be skeptical that such data exists.
Collecting data on the ethnicity of individuals or business owners in the U.S. is a touchy subject - it's generally only done for census purposes (and then released only in aggregate) or to judge compliance with equal-opportunity laws. In the latter case, any data released would only be in aggregate, as well.
You should be able to get information on which cities or neighborhoods have high concentrations of people with an Asian background, but I doubt very much you are going to get any kind of list of "Asian businesses". Try www.census.gov
Best bet, then, would be trade associations, which may or may not make this data available for free.
In many countries, this would be a copyright violation, and these directories have intentional mistakes to identify violators.
The subject has been discussed here before (do a search for "yellow pages") and the consensus seems to be that in the U.S. at least, yellow pages aren't protected. But is a specialized directory the same as a general yellow pages directory?
I would think that the more specialized the directory, the more claim they have to copyright protection.
In any case, in the U.S. you are not going to get this data from "the public record".
I guess you could get business names and owners and scan for words in the business name and names of business owners, which might be a close approximation. But getting the data would not be easy - there is no central, national registry of business names.
Incorporated businesses are handled by each of the 50 states.
Assumed names are filed locally - typically at the city or county level.
Neither of the above identify "Asian" businesses as such.