menu for custom type and taxonomy at the same time or filter by type

I have two custom types (eg: Cooks and Restaurants) and both have a shared taxonomy (eg: specialty). I want a menu that pulls all the restaurants based on a specific specialty but there doesn’t appear to be a way to use appearance -> menu to add the specialty taxonomy so that it only links to restaurants but not to cooks. eg: a menu that basically does a search for one type with that taxonomy match but doesn’t also pull the other type with that taxonomy match

To be clear I can get the taxonomy with both by simply editing a menu, placing a check-mark in the taxonomy name in screen options and then adding the individual taxonomy terms… I just don’t see how to filter the results to be only a single type. I assume that I will need to do this with some code but wanted to be sure I wasn’t over-engineering it.

Is there a way that is built into WordPress that would allow this to be done or am I looking at writing some custom code/adding a plugin?
Alternately… what keywords should I have used to find an answer to this? It seems like anything that is remotely related is dealing with using a taxonomy with the same name as a type. I also looked at the title (at a minimum) of all the advanced-taxonomy-queries tagged questions when I discovered that but none seemed to be like my issue (though many were interesting in other ways).