Slice the eggplant into 4 slices. Place in a colander in the sink and sprinkle with 1 teaspoon salt. Let sit 5 minutes.
Slice the chicken breasts in two. Season with remaining salt and pepper.
In a cast iron skillet, mix breadcrumbs and garlic. Place over the grill, watching constantly, until the breadcrumbs are golden brown and toasted (5 minutes). Set aside.
Grill the chicken about 10 minutes on each side, until fully cooked.
While the chicken is cooking, grill the eggplant about 5 minutes on each side until fully cooked.
In the breadcrumb skillet on the grill, cook the tomatoes until slightly wilted.
To make a stacker, place one slice of eggplant down. Top with cheese. Top with two slices of tomato. Top with chicken. Top with 2 slices of tomato. Repeat with the second slice of eggplant, cheese, tomato, chicken, ending with tomato. Top with 1/4 cup breadcrumbs.
Repeat for remaining eggplant and chicken. Serve immediately.