Ad Astra Food and Drink
Per a review on Yelp:
Driving out through the Flint Hills is pleasure enough, but Ad Astra makes it extra special. This quaint little restaurant has become so popular that it is taking reservations and obviously gets patrons from Emporia, Council Grove, and points all around. It's right near the Tallgrass Prairie Preserve, too.
The main thing here is that all of the food is house-made and nearly all locally-sourced. If you think that in Strong City you'd be getting a menu featuring strictly steaks and chicken fingers, well, you'd be wrong; the veggie burger is also made in-house and is excellent. There's good beer on tap, a small wine list, very friendly people, a smartly-appointed dining room, and real ambiance.
On your trek think about calling ahead first to be sure you get a table--if you have to wait there won't be much else of an alternative! (That's part of the charm of exploring the Flint Hills in the first place, though.)