Heat the oil in a large saucepan, and fry the mushrooms on a medium heat until browned. Remove from the pan.
Melt the butter in the same pan, and fry the onions and celery for 4-5 minutes until softened. Add the flour to the pan, and stir together using a whisk, until the flour is incorporated with the vegetables and butter. Turn the heat up to high, and pour in the milk in a slow stream, whilst stirring with the whisk, until the milk is completely mixed in.
Add the rice and white wine, and stir. Then add in the stock and dried thyme and stir again. Bring to gentle bubble and simmer for 20 minutes – stirring 2 or 3 times during cooking. Test the rice – it should be cooked at this point (give it a couple more minutes if it needs it).
Add in the salt, pepper, lemon juice, cream, cheese, cooked chicken and the mushrooms. Stir together and test for seasoning. If you prefer your soup a little thinner, you can stir in a splash of milk.
Heat until the chicken is warmed through, then serve topped with fresh parsley.