Cut four 1/3-inch-thick lengthwise slices from the eggplant (discard the 2 end slices). Place the eggplant slices in a colander in the sink, sprinkle lightly with salt and let stand for 15 minutes; pat the eggplant dry.
Meanwhile, in a food processor, pulse the torn bread to form large, coarse crumbs; measure out 1 1/2 cups. In a large, heavy skillet, heat 2 tbsp. EVOO over medium heat. Add the breadcrumbs and cook, stirring occasionally, until nicely toasted, 6 to 8 minutes. Transfer the crumbs to a small bowl and wipe out the pan.
In the reserved skillet, heat 3 tbsp. EVOO over medium heat. Add the eggplant and cook, turning once or twice, until golden and tender, 6 to 7 minutes. Transfer the eggplant to a plate and evenly spread one-fourth (about 1 1/2 tbsp.) of the goat cheese on each slice; loosely cover with foil to keep warm.
Return the pan to medium-high heat, add the remaining 2 tbsp. EVOO and swirl to coat. Lightly season the chicken with salt and pepper. Add to the skillet smooth side down and cook over medium-high heat, turning once or twice, until just firm to the touch, 7 to 8 minutes. Transfer to serving plates and arrange a cheese-covered eggplant slice on top of each.
Add the tomato slices to the skillet and cook, turning once, to warm through, about 2 minutes. Place a slice on each chicken cutlet.
Add the diced tomato and 1/2 cup water to the pan, scraping up the browned bits, and bring to a boil. Cook, stirring, until thickened, about 3 minutes. Season lightly with salt and generously with pepper. Spoon the sauce over the chicken and sprinkle with the breadcrumbs.