In a bowl, combine cabbage, carrot, mint, cilantro, and chile. Stir in lime juice and olive oil and season to taste with salt.
In a pie plate mix flour, cornmeal, 1 1/2 tsp. salt, and cayenne. In a medium bowl, whisk buttermilk and egg to combine. Dredge chicken in flour mixture, dip into buttermilk mixture, then return to flour mixture, turning to coat all sides.
Pour 1 in. of oil into a 5- or 6-qt. pan over medium-high heat. When oil reaches 350°, add chicken in a single layer (in batches if necessary), and cook until well browned all over and no longer pink in the center (cut to test), about 6 minutes per batch. As chicken is done, transfer to paper towels and sprinkle lightly with salt.
Spread sandwich rolls with mayonnaise. Divide chicken among rolls and top generously with slaw.
Wine note: Typically full of honeyed tangerine, peach, and honeysuckle, Viognier--the great white grown in France's Rhône Valley--is rich but, at its best, crisp and minerally too. It pairs well with creamy sauces and aromatic spices. Leaner versions stand up to spicy Thai and Vietnamese flavors; richer ones are good matches for nuts, cheese, mild Indian curries, and sweet shellfish.
Alban Viognier 2005 (Central Coast; $23). An orchard full of peach-blossom aromas followed by orange-honey flavors wrap around the chicken sandwiches.
Clos LaChance Estate Viognier 2005 (Central Coast; $20). A peachy, citrusy minerality--call it wet stones--picks up on the mint and cilantro in the slaw on the sandwiches.
Note: Nutritional analysis is per sandwich.