Heat a large pan over low heat, then add the olive oil and onions. Cook for 3 to 5 minutes or until the onions soften.
Adjust the heat to medium, then add the mushrooms. Cook for 3 to 5 minutes until the mushrooms begin to shrink. Add the garlic and thyme, and cook for 30 more seconds.
Add the rice, and cook until it begins to toast and make a crackling sound. Pour in the wine, and stir to combine. Add just enough broth to cover the rice.
Simmer, stir often, and continue to cook until the rice is almost dry, about 10 minutes. Add more broth to cover the rice. Repeat until the rice is soft on the outside but undercooked in the center.
Add the frozen peas and more broth to cover the rice. Cook at a simmer for an additional 10 to 12 minutes stirring often. The rice should be al dente.
Add a little more broth, about 1 to 1 ½ cups, along with Parmesan cheese.
Stir to combine, and taste. Season with salt and pepper.
Serve hot with fresh bread or a salad.
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