Cut the eggplant into ½-inch rounds and place in a single layer on a cutting board covered with 2–3 sheets of paper towels. Lightly season both sides of the eggplant with salt and allow the eggplant to release its bitter liquid, for about 10 minutes.
Pulse pine nuts in a food processor until they are completely broken down. Scrape down the sides of the bowl with a spatula to ensure that the nuts have been fully processed. Add olive oil and garlic and pulse until the garlic is finely chopped.
Add basil leaves to the food processor and pulse, stopping every 10–15 seconds to push the leaves down. Blend until the basil is entirely worked into the olive oil. Add the lemon juice and pulse for a few seconds more, to prevent the basil from turning brown. Adjust for salt and pepper and cover with plastic wrap. Reserve.
Slice the tomatoes and mozzarella cheese into ¼-inch-thick slices. Lightly season the tomatoes with salt. Set aside.
Pat the eggplant rounds dry, then brush both sides of each round with olive oil.
Heat a grill pan over medium heat or preheat a grill to medium. Transfer the eggplant rounds to the grill pan and cook. Grill the eggplant rounds until they are tender and grill marks appear, 3–4 minutes. Flip the rounds and continue to cook for an additional 3–4 minutes. Remove from the heat and set aside.
Alternate layers, starting with eggplant rounds, sliced tomato, and mozzarella. Repeat this order one more time. Finish by lightly drizzling the pesto dressing over the top of each stack and garnish with a fresh basil leaf.