Grilled method: Preheat a grill to medium high heat (375 to 450 degrees Fahrenheit). Make sure the grates are clean. Right before grilling, brush each side with olive oil. Sprinkle the tops with kosher salt and fresh ground pepper. Grill for 10 to 20 minutes (depending on the grill), turning halfway through, until tender and grill marks form. Alternatively, you can use a grill pan on the stovetop: heat the pan and grill the eggplant slices until they are tender and grill marks form.
Roasted method: Or, you can roast the eggplant! Go to Perfect Roasted Eggplant and follow the instructions for eggplant rounds.
Toast the bread: Toast the bread in the remaining minutes on the grill, or place it in the oven for a few minutes to toast it.
Assemble the sandwiches: Thinly slice the tomatoes. Tear the mozzarella into pieces. Spread pesto on the cut side of one piece of bread, and olive spread on the cut side of the other bread. Place grilled eggplant on the bread, and top with mozzarella, sliced tomatoes, and basil, if using. Top with the other bread slice. If desired, wrap in aluminum foil to help contain the mess when eating!
Make ahead notes: The sandwiches can be made a few hours in advance for picnics. If doing so, you may want to slice the tomatoes and let them sit on a cutting board for several minutes to drain excess liquid.