This is the result from OSH Park for my nested rings (see below).
To reiterate, I created each ring as a separate KiCAD project and then opened PCBNew manually (outside of KiCAD) to append both boards to the same file for Gerber export.
OSH Park recommended at least a 100mil gap between the two boards (enough for routing) and that in that gap, I add the words "Cut Out".
edit: I did not create the mouse-bites or the pattern that the fab routed, I just created a space between the two rings - they figured out how to route it out and where to put the mouse-bites. I don't know if OSH Park did that during panelization (probably) or the fab.
The result is just perfect for what I need and allowed me to make more efficient use of the space - pulling two boards in the same space as one. Arguably, I could put a third board on the inside to make use of that space as well but, honestly, I didn't have anything that would fit that space.