Preheat the oven to 400 degrees . On two parchment-lined baking sheets, arrange the eggplant and potato slices in a single layer. Brush both sides with 2 tbsp. oil; season with salt. Bake until tender, turning halfway through, about 40 minutes.
Meanwhile, heat 1 1/2 tsp. oil in a large nonstick skillet over medium. Add the chicken and onion. Cook, stirring often, until the chicken is just cooked through, about 8 minutes. Add the garlic; stir 1 minute. Add the tomatoes, vinegar and cinnamon; season. Cook, stirring often, until the tomatoes are soft, about 8 minutes.
In an 8-inch square baking pan, arrange half the potato slices in a single layer; top each potato slice with an eggplant slice. Spoon half the chicken mixture on top of the eggplant, creating individual stacks. Repeat the layering with the remaining potatoes, eggplant and chicken mixture. Bake until heated through, about 10 minutes. Let rest 5 minutes.
Meanwhile, in a small bowl, whisk the yogurt and remaining 1 1/2 tsp. oil; season.
Divide the vegetable stacks among plates; drizzle with the yogurt sauce.