Preheat the oven to 220°C (200°C fan). In a large pot, bring enough salted water to a boil for the pasta. Precook the pasta for ca. 6min. until very al dente. Meanwhile, peel, halve and finely chop the onions. Scrub or peel the carrots, then coarsely grate them. Cut the tomatoes into small cubes. Reserve a cup of the cooking water and drain the pasta in a sieve.
Heat 1Tbsp of olive oil in a large pan over medium-high heat. Add the meat, breaking it apart with a spatula. Season with a large 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 the spice mix and 2Tbsp 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 150ml of the reserved pasta cooking water. Stir well to combine. Remove from the heat. Whisk 1 egg and stir it into the béchamel 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.