Warm the stock in a saucepan over low heat. Heat 2 tablespoons of the oil in a wide, heavy pan over medium-high heat. Add the mushrooms in a single layer and cook without stirring for 3 minutes, then stir and keep cooking until browned and their liquid has evaporated, 6–8 minutes more. Season with a pinch of salt, add the thyme and 1 tablespoon butter, then transfer to a bowl.
Lower the heat to medium. Add the remaining oil and the shallot and cook until soft, about 3 minutes. Stir in the garlic for 30 seconds.
Add the rice and stir for 2 minutes, until the grains are coated and the edges look translucent.
Pour in the wine and stir until it is almost completely absorbed.
Add a ladle of warm stock and stir until it is mostly absorbed. Continue adding stock one ladle at a time, stirring frequently, for 18–22 minutes, until the rice is creamy and tender with a slight bite.
Stir in most of the mushrooms, the parmesan, the remaining butter and the salt. Add one more splash of stock so the risotto is loose, cover and rest for 2 minutes. Serve topped with the reserved mushrooms, parsley and extra cheese.