Make sure your knife is sharp before starting. Cut the end off the eggplant then cut it int thin slices, around 0.1in/0.25cm thick lengthwise. Discard the smaller pieces that are mainly skin and not as long from either side.
Slice the mozzarella and tomato very thinly as well. Shred the basil leaves thinly.
Warm a griddle pan and lightly brush the eggplant slices with olive oil. Alternatively, drizzle on a little and quickly rub over before it is absorbed. Place the eggplant slices on the pan and grill for a couple minutes each side . They should soften and have light grill marks. As the second side is almost done, add a larger piece of mozzarella in the thick part of the eggplant slice. Top it with a slice of tomato, and add a smaller piece of mozzarella at the thiner end. Sprinkle over a couple pieces of basil and drizzle a little olive oil and a couple grinds of black pepper, if you like. Let it cook fir a minute more then carefully remove from the pan. There will be a bit of liquid comes out from the tomato and cheese so let it drain off.
Roll the eggplant from the thiner end, which has only the cheese. You probably won't get it to roll completely, but once it's close, hold together with a cocktail stick. Serve warm (better) or room temperature.