Blend flour, salt, chilled butter, and shortening in a food processor. Gradually add ice-cold water to it while blitzing until crumbs form.
Turn it out onto a lightly floured work surface. Quickly shape the dough into a disk, and wrap it with plastic wrap. Refrigerate it for at least 2 hours.
Preheat the oven to 375°F.
In a pan, cook onion and garlic with olive oil for 1 minute over medium heat.
Add the chicken and cook for another 2 minutes. Stir occasionally.
Add potatoes, mushrooms, half the amount of salt, pepper, and thyme. Stir well and allow to cook for another 10 minutes.
Whisk eggs, milk, cream, and the remaining salt in a bowl.
Unwrap the dough, and roll it out thinly on a work surface. It can stick to your rolling pin, so be sure to flour your work surface and the rolling pin.
Gently roll the dough sheet around the rolling pin, then lift it from the work surface. Unroll the pastry over a round baking dish, very carefully. Gently press the pastry on the sides of the dish to form the crust.
Transfer the cooked filling into the crust. Slowly pour the egg-milk mixture on the filling and swirl it around using a spatula.
Bake at 375°F for 30 minutes or until the top is golden. Check every once in a while.
Sprinkle cheese on top of the quiche and bake for another 5 minutes.
Sprinkle parsley on top and serve.