The usage has no currency among native Anglophones, but a number of people have asked about it online. And I'm pretty sure they're all asking about the same source (Crying in H Mart by Michelle Zauner).
I picked these three posts out from the relevant Reddit page...
The phrase: “she has a salty hand” means to be a bit careful, not timid, but thinking of possible over-indulgence. You can have a “sweet hand” which would be an opposite.
Even with the context, it’s basically impossible to guess what “salty hand” means here.
We can tell that English is not the mother’s native language, and the author makes a point of explaining how she misspeaks and calls steaming hot “steamy hot.” [italics mine]
My guess is that “salty hand” is another misspoken expression. Maybe it’s apparent later what it means, maybe not.
Since the author was born in Korea and raised in Oregon by a Korean mother and American father, I suspect that "a salty hand" might be a hybrid-English phrase stemming from the expression "a heavy hand with the salt." If she heard this growing up, she might not even realize it is not a standard English expression, especially if it's use was re-enforced by other members of a non-native community.