"Not-to-miss"?? No. Not really. In Jacksonville, Hooters counts as fine dining and culture. But I've had good meals at the following places (5-7 years ago):
The Wine Cellar in Jacksonville. Probably the most wine-serious place.
Wine Cellar The Grill in the Ritz Carlton of Amelia Island. Typical RC restaurant. Fine food and wine. Pricey, and no soul or passion.
Beech Street Grill in Fernadina Beach near Amelia Island. Best of the lot. Nicely appointed in an old house in a cute old town. Expressive menu and good wine list.
Beech Street Grill Good Luck.