Thanks for the suggestions so far.
The more I look into it, I feel like it definitely is not an Italian-only thing. This grandma just happens to be Italian.
The sort of cookies I'm talking about vary in shape, size, form, flavor, etc, but they are all very small and have what I can only assume is a large amount of butter. They are very crunchy and rich - not puffy, soft, bready, or dry at all.
In addition to Palermo, it looks like Original Ferrara in Little Italy may have what I'm after. I'm also not opposed to Dinkel's, as I remember liking their butter cookies. I think a taste test is in order!
Any other specific bakers to hit, please let me know.
I feel like these sort of cookies are a lot like biscuits. A lot of places make them, but only a few make any really worth eating.