To make the meat sauce:
Fry the lamb mince in a large deep pan until cooked through. Drain most of the oil from the pan if you dislike it. Add chopped green spring onions and a finely diced carrot (as well as grated courgette), altogether with paprika, cinnamon, thyme and rosemary and sugar. Mix in well and fry for a few minutes and add a Low FODMAP beef stock. Cook until the carrots are soft. Then add the tomato passata, balsamic vinegar and tomato puree. The reason for this is because acid from the tomatoes often stops the hard vegetables from cooking until soft and they stay crunchy. Cook the sauce for 30 minutes, until lamb is even more tender and the sauce has thickened.
Preparing the potatoes and aubergine:
In the meantime, when the sauce is piping away:
Cook the sliced potatoes in a boiling water, until nearly tender, but still hold their shape, then drain under cold water until cold.
Slice the aubergine lengthwise into thin slices. Spray a hot griddle pan with oil and grill the aubergine slices on both sides, until they have a nice grill marks.
Preheat the oven to 200 C/ 180 Fan.
To make the cheese sauce:
Start heating up the milk, reserving few tablespoons in a small bowl with the cornflour. Stir in the cornflour with the milk until the cornflour is dissolved. Pour the cornflour into the milk, and while the milk is heating up, keep whisking into the milk.
When thickened, add the parmesan, grated small, into the milky mixture, together with seasonings, including a tiny pinch of ground fresh nutmeg and whisk again. Let it cool.
In the end, when the sauce is cooled, stir in the beaten egg.
To assemble:
Place a one-third of the meat sauce into a large ovenproof dish.
Cover with a third of the potatoes and then a third of the grilled aubergines. Repeat the process twice more, having aubergines as the last layer. Pour the white sauce evenly, making sure it covers everything. Grate the remaining parmesan on top.
Bake for 35-45 minutes, until bubbling away and golden brown.