Pour vegetable oil into a skillet to a depth of about 1/4 inch and heat on high heat until hot.
Break the egg into a bowl and whisk. Pour the cornmeal into a second bowl, add the cornstarch and some salt and pepper and mix.
Dip the tomato slices first in the egg and then in the cornmeal mixture to coat completely. Add to the skillet and fry for 1 to 2 minutes a side.
Spread both slices of bread with the mayo and sprinkle with salt and pepper. Pile the cooked tomatoes onto one of the bread slices. Place the basil leaves onto the second slice, then sandwich the slices. Cut in half and serve.