Preheat a grill to medium-high. Place the eggplant slices in a large bowl and toss with the olive oil and balsamic vinegar. Season with salt and pepper. Grill the eggplant slices, flipping once, until they acquire nice grill marks on both sides, about 6-8 minutes total. Transfer the slices to a baking sheet and cover tightly with foil to allow the eggplant to steam. (You can also cook the eggplant in the oven. Line the slices on baking sheets, overlapping slightly, and bake in a 425˚F oven for 15 minutes, or until tender.) Do Ahead: The eggplants can be cooked several hours in advance and left at room temperature, or refrigerated overnight.

Preheat the oven to 425˚F with a rack in the upper third.

In a large skillet, melt the butter over medium-high heat. Add the onion, and cook until tender, about 4 minutes. Stir in the garlic and cook 30 seconds or so, until fragrant. Add the ground beef and season with salt and pepper. Cook, stirring occasionally and breaking up any chunks, until the meat is browned, about 5 minutes. Sprinkle in the cinnamon and let cook 30 seconds. Pour in the wine and simmer until the liquid is nearly evaporated (3-4 minutes). Add the diced tomatoes, and season with more salt and pepper. Simmer until the tomatoes are softened, about 3 minutes. Taste and add more salt and pepper as needed.

Lightly oil a 13x9-inch baking dish. Spread a scant cup of the meat sauce in the bottom of the pan. Layer half of the eggplant slices over the sauce. Pour half of the meat sauce over top and spread in an even layer. Sprinkle with half of the feta and three-quarters of the mint. Layer the rest of the eggplant slices on top, followed by the remaining meat sauce and feta. Bake on a rack in the upper third of the oven for 10-15 minutes, or until lightly browned. Let stand 5-10 minutes. Serve warm, garnished with the remaining mint. Do Ahead: The dish can be baked up to 2 hours in advance and left at room temperature. Reheat slightly before serving. It’s also delicious the next day.