Pour the oil into a wide frying pan - it should come about 0.5cm up the sides - and place over a high heat. When hot, fry the aubergine in batches for a minute a side, until nicely tanned. Transfer to paper towel. Repeat with the potato and then the cauliflower and carrot, but for only about 30 seconds a side - these vegetables need to take on some colour but stay crunchy.
Cut out a circle of greaseproof paper large enough to cover the base and some of the pan's edge, and line the pan with it. Cover the paper with aubergine slices, then layer up, in order, with the carrots, potatoes, tomatoes and cauliflower. Sprinkle the garlic on top and cover with rice.
Mix the boiling water or stock with the spices and seasoning. Gently pour the mix over the rice, making sure all the rice is immersed. Put the pot on the stove top and bring gently to a boil (you don't want a vigorous simmer because that will ruin the layers). Once simmering, reduce the heat to a bare minimum, cover the pot and cook for 30 minutes. Remove from the heat, lift off the lid, place a clean tea towel over the pan, pop the lid back on and leave to rest for 10-15 minutes.
While you wait for the rice to cook, make the salad. In a bowl, whisk the tahini paste, water, lemon juice, garlic, dried mint and parsley. The mix should be creamy but still runny enough to pour. If it is too thick, add more water. Add salt to taste - be generous. Put the tomato and cucumber chunks in a bowl, pour the dressing on top, mix gently, drizzle with oil and sprinkle with fresh mint. Set aside.
When ready to serve, remove the lid and towel, and place your large serving plate over the pan. Carefully turn over, so the plate is now on the bottom and the pan on top, place on a stable surface and gently lift off the upturned pan. Carefully lift off the greaseproof paper - ideally, you want the rice to keep the pan's shape, though it's not the end of the world if it falls apart. Serve hot or warm with the salad on the side.