Private, intimate lunch spots in/ around the loop?

I’m looking for lunch spots in the Houston area that would be good to go to for an intimate and private conversation.

Oak Room at the Post Oak is a perfect example of the vibe/style I’m looking for, minus the membership requirement. Just a simple restaurant with some aspects of privacy type of thing, to converse privately.

To clarify, I’m not looking for anything particularly as fancy as the Oak Room either.

Thanks in advance!

Edit: typo