Heat half the fat (75 g) in a large ovenproof pot or Dutch oven over medium-high heat. Add the lamb and braise, stirring occasionally, for 8-10 minutes until browned on the edges. Season with salt and pepper.
While the lamb browns, prepare the vegetables: peel and dice the 3 onions and 2 bell peppers, peel and cube the potatoes, cut the 2 zucchini and 1 eggplant into cubes, trim the green beans, roughly chop the tomatoes, and finely chop the parsley.
Add the remaining fat (75 g) to the pot with the lamb and heat over medium. Stir in the diced onions and sauté for 3-4 minutes until softened. Add the bell peppers, potatoes, zucchini, eggplant, and green beans. Sauté for 5-6 minutes, stirring occasionally. Season with salt, add the chopped parsley and tomatoes, and stir well.
Pour in 1 cup of water, cover the pot, and bake at 180°C (350°F) for 45-50 minutes until the lamb is tender and the vegetables are cooked through.
Whisk together the 2 eggs and 3 tablespoons yogurt until combined. Pour the mixture over the stew and return the uncovered pot to the oven at 180°C (350°F) for 5-7 minutes until the topping is set and lightly golden.