Preheat your oven to 400°F (200°C). Brush the sliced eggplants with olive oil, then sprinkle them with salt and pepper. Arrange the slices in a single layer on a baking sheet and roast for about 20 minutes, until tender and starting to brown.

In a large skillet, heat a splash of olive oil over medium heat. Sauté the chopped onion and minced garlic for about 3-4 minutes, until fragrant. Add the browned ground beef and cook for another 5 minutes. Stir in crushed tomatoes, tomato paste, beef broth, cinnamon, oregano, and season with salt and pepper. Simmer for 10-15 minutes until it thickens.

In a saucepan over medium heat, melt 3 tablespoons of butter. Whisk in flour, cooking for about 1-2 minutes. Gradually pour in milk while whisking. Cook until the mixture thickens, then remove from heat. Allow it to cool briefly before whisking in the egg and grated cheese.

In a greased baking dish, layer half of the roasted eggplant slices on the bottom. Spoon all of the thickened meat sauce over the eggplant layer. Follow with the remaining eggplant slices, and finally pour the béchamel sauce over the top.

Preheat your oven to 350°F (180°C). Bake the assembled dish for 40-45 minutes, or until the top is bubbly and golden brown. Let it rest for about 15 minutes before serving.