For the basil mayonnaise: Pulse the basil, mayonnaise, lemon juice and mustard in a food processor until smooth. Season with salt and pepper. Cover and refrigerate until needed.
For the fried tomatoes: Season the tomatoes with salt and pepper. Place the flour in a shallow dish. Whisk together the buttermilk and hot sauce in another shallow dish. Whisk together the panko and cornmeal in a third shallow dish. Working with 1 slice of tomato at a time, coat the tomato first in the flour, then in the buttermilk and then finally in the cornmeal. Repeat this process with the remaining slices.
Add enough vegetable oil to a large skillet to coat the bottom and heat over medium to medium-high heat. Working in batches, fry the tomato slices until golden brown, about 2 minutes per side. Drain on paper towels. Serve the tomatoes warm with the basil mayonnaise.