1. In a medium shallow bowl, combine the cornmeal, salt, black pepper, and cayenne pepper; set aside. In another shallow bowl, whisk together the buttermilk, eggs, and lime juice (the mixture may appear curdled); set aside.

2. In a large cast-iron or heavy skillet, pour oil to a depth of 1/2 inch; heat until a deep-fry thermometer measures 375 degrees. Meanwhile, working in batches, dip tomatoes in buttermilk mixture, then in cornmeal mixture. Set aside on a large plate; repeat with remaining tomatoes.

3. Fry tomatoes, working in batches, until golden brown, about 1 minute on each side. Drain on a paper-towel-lined plate. Season with salt while hot. Serve warm with basil-lime mayonnaise.