Preheat a grill pan or countertop grill to high. Place a baking sheet in the oven and preheat to 250 degrees . Combine the vinegar and 2 tablespoons EVOO on a plate and coat the chicken with it. Sprinkle with the rosemary, season with salt and pepper and set aside.
Pour 1/4 cup EVOO into a small dish. Using a pastry brush, paint the eggplant slices on both sides with the oil; season with salt and pepper.
Grill the chicken for 2 to 3 minutes on each side, then place in the oven to keep warm. Grill the eggplant until tender, about 10 minutes. Add to the chicken.
While the chicken and eggplant cook, heat the remaining 2 tablespoons EVOO, 2 turns of the pan, over medium heat. Add the garlic, onion and crushed red pepper; cook until the onion is tender, 6 to 7 minutes. Add the tomatoes and heat through; season with salt and pepper.
Preheat the broiler to high. Remove the chicken and eggplant from the oven. Assemble the 4 stacks on the baking sheet by layering a chicken cutlet, a slice of eggplant, a little sauce and a slice of smoked mozzarella, a second chicken cutlet, another eggplant slice, more sauce and another slice of mozzarella. Broil the stacks until the top layer of mozzarella melts. Sprinkle the basil on top and serve the remaining sauce on the side.