Preheat the oven to 180C and line a roasting tray with baking paper.
Place the chicken breast on the tray,rub in the olive oil and mint,and drape a few lemon slices on top. Roast for 15 to 20 minutes or until cooked (insert a skewer into the thickest part of the breast and check that the liquid runs clear;if using a regular chicken breast reduce the cooking time by five minutes and check regularly). Using a few forks,shred the meat,skin and lemon into a bowl. Set aside while you prepare the soup.
Pour the stock into a small saucepan and bring to the boil. Meanwhile,add two tablespoons of olive oil to a frypan over medium heat. Add the burghul and lemon zest to the frypan and cook,stirring regularly,until the burghul begins to colour and has a nutty aroma.
Ladle the hot stock over the burghul (you want to just cover the grains) and cook for five to 10 minutes or until the grain is soft and the liquid has been absorbed (you may need to add more stock depending on how quickly the stock evaporates). Add the burghul to the bowl of shredded chicken,then add the cooked rice,garlic and parsley. Season generously and stir to combine.
To make the chilli oil,combine the hot sauce and olive oil in a small bowl and set aside.
Add the stock and yoghurt to a large saucepan over low heat. Bring the heat up very slowly so that the yoghurt doesn't split and curdle. Cook,stirring regularly,for five minutes or until the soup is hot. Crack in the egg and sift over the cornflour,stirring continuously until fully incorporated. Once the soup has thickened slightly,it is ready. Stir through the dried mint and season with salt and pepper.
To serve,divide the chicken and burghul mixture between bowls. Carefully ladle the soup mixture and spoon over some of the chilli oil. Add the strips of lemon peel and serve immediately.