Heat oven to 425°F (220°C). Line two rimmed sheets with parchment. Toss eggplant and potatoes with the 3 tablespoons oil, salt and oregano. Spread separately without crowding. Roast 30–35 minutes, turning after about 20 minutes, until the potatoes are tender and the eggplant is soft and browned. Use more time if needed.

Meanwhile, heat 1 tablespoon oil in a large skillet over medium heat. Cook onion about 5 minutes until softened. Add garlic for 30 seconds, then beef. Break it into small pieces and cook thoroughly; ground beef should reach at least 160°F (71°C). Spoon off excess rendered fat if needed.

Stir in tomato paste, crushed tomatoes, cinnamon, oregano, salt and pepper. Simmer uncovered 15–20 minutes, stirring, until thick enough that a spoon leaves a brief trail and no loose puddle forms. Add a splash of water only if it catches before thickening.

Melt butter in a saucepan over medium-low heat. Whisk in flour and cook 1 minute without browning. Gradually whisk in warm milk, keeping the mixture smooth. Bring to gentle bubbles and cook 3–5 minutes, stirring, until it coats a spoon. Remove from heat; stir in 40 g Parmesan, nutmeg and salt.

Reduce oven to 375°F (190°C). Lightly oil the baking dish using any oil remaining on the vegetable trays. Arrange potatoes across the base. Add half the eggplant, half the meat sauce, remaining eggplant and remaining meat sauce. Spread béchamel over the top and sprinkle with the remaining 20 g Parmesan.

Bake 35–40 minutes, until the top is golden and the center reaches 165°F (74°C). If the top colors too quickly, cover loosely with foil. Rest about 30 minutes before cutting into 8 portions. Add optional parsley. Refrigerate leftovers within 2 hours in shallow covered containers and use within 3–4 days; reheat to 165°F (74°C).