Melt the butter over medium high heat.
In a separate, small pan, heat the chicken or vegetable stock over medium heat.
As soon as the butter melts, add the onions and stir to coat. Cook for 2 minutes, do not let burn.
Add the rice and stir to coat. Let the grains toast for 1 minute.
Pour in the white wine and stir. Let the wine mostly absorb into the rice, this will take a few minutes. Once you can use the stirring spoon to make a part in the rice, add a cup of warm stock and stir in. Add the mushrooms at this point too.
Let the rice mixture simmer over medium high heat, stirring occasionally, and add another cup of stock until it absorbs each time, until you get to the last one. It should have been absorbing liquid for 15-20 minutes by this point.
Once the last cup of stock has halfway absorbed into the rice, about 3 minutes, remove from heat. Stir in thyme, salt, pepper, and cheese. Season to taste and serve immediately.