For the tomatoes: In a large heavy-bottomed saucepan, pour enough oil to fill the pan about a third of the way. Heat over medium heat until a deep-frying thermometer inserted in the oil reaches 350 degrees F.
In a shallow dish, add the cornmeal, paprika, salt, cayenne and pepper. Mix will. Place the flour on a separate shallow dish. Place the beaten eggs into another shallow dish. Dredge the slices of tomato into the flour, then the eggs and finally in the seasoned cornmeal. Place the breaded tomatoes in the oil and fry until golden brown, about 3 minutes on each side. Remove and drain the tomatoes on paper towels. Season with salt and pepper.
For the sauce: Mix the mayonnaise, buttermilk, barbecue sauce, vinegar, lime juice and brown sugar in a bowl. Season with salt and pepper. Serve the sauce alongside the fried tomatoes.