Another vote for Mt Pisgah Inn!
Love that place. Depending on how much time you have, find a clear day and go out on the Devils Courthouse. Then, drive down to Brevard on 278, stopping at the Cradle of Forestry, Looking Glass Falls, Sliding Rock (if you are brave and it is warm), the Fish Hatchery (great place for a picnic), and then Brevard. Beautiful town.
You’ll also want to stop at Mt Mitchell, Grandfather Mountain, enjoy the drive over the Linville Viaduct, the Peaks of Otter, and a BUNCH of overpasses..
Make that Overlooks.
I’ll second Sliding Rock...it’s a blast!