"She's like a cockroach what turned into a butterfly."

This is a line from Pocketful of Miracles (1961)

She’s like a cockroach what turned into a butterfly.

Apparently in "standard English grammar" this should be

She’s like a cockroach that/which/who turned into a butterfly.

Presumably this usage of "what" in place of "that" is dialectal. What kind of dialect has this grammar phenomenon? What is this grammar phenomenon called in linguistics? I recall seeing a comment by MetaEd hinting at this, but they didn’t specify. The movie is an American movie and I believe MetaEd is an AmE speaker as well. This Q&A seems to suggest this occurs in BrE but I am also curious about its currency in AmE.

I guess you have no acquaintances what speak dialect. – MetaEd Mar 13 ’13 at 3:30