Den Backofen auf 220°C (200°C Umluft) vorheizen. In einem mittelgroßen Topf ausreichend gesalzenes Wasser für die Pasta zum Kochen bringen. Die Pasta in das kochende Wasser geben und ca. 6Min. kochen. Mit einer Tasse etwas Pastawasser abschöpfen, dann die Pasta in ein Sieb abgießen und abtropfen lassen. Die Karotte ggf. schälen und grob raspeln. Die Tomate grob würfeln.
Heat ½Tbsp of olive oil in a medium pan over medium-high heat. Add the meat, breaking it apart with a spatula. Season with a pinch of salt and pepper and cook for 4–5min. without moving it too much, allowing the meat to develop a caramelised crust.
Add the onion, carrot and tomato to the pan and sauté for ca. 5min. until the vegetables soften. Then stir in ½ the spice mix and 1Tbsp balsamic vinegar and cook for ca. 30sec. until the spices are fragrant. Season with salt and pepper to taste.
Pour the passata into the pan and add about 75ml of the reserved pasta cooking water. Stir well to combine. Remove from the heat. Whisk 1 egg, then mix ½ of the egg into the béchamel sauce. Tip: If you want, you can add the rest of the egg into the tomato sauce.
Add the pre-cooked pasta into a deep baking dish and press to a firm layer. Distribute the tomato sauce evenly on top. Spoon the béchamel over the top, using the back of your spoon to spread it into an even layer that covers the entire dish. Scatter the cheese over the top of the béchamel layer.
Transfer the baking dish to the oven and bake for 12–14min., or until the top is bubbling and has developed some golden-brown spots. Tip: Add the broiler/grill function of the oven during the final 5min. Remove the pastitsio from the oven and let rest a few minutes before serving.