In a medium saucepan over medium heat, bring the broth to a gentle simmer. Once simmering, reduce the heat to low to keep it warm.

Heat the olive oil in a large pot or Dutch oven over medium heat. Add the chopped onion and cook, stirring frequently, until it turns translucent, roughly 5 minutes. Introduce 1 tablespoon of butter, the minced garlic, sliced mushrooms, bay leaf, and thyme leaves. Sauté until the mushrooms soften and turn golden, about 4 minutes. Season with salt and pepper to taste, then transfer the mixture to a bowl and set aside.

In the same pot, melt the remaining tablespoon of butter. Add the arborio rice, stirring continuously to ensure the grains are evenly coated. Cook until the rice emits a nutty aroma, approximately 2 minutes. Pour in the white wine and cook, stirring, until it is mostly absorbed by the rice.

Using a ladle, add about 1 cup of the hot broth to the rice. Stirring frequently, cook until the liquid is mostly absorbed. Continue adding the broth, one cup at a time, allowing the rice to absorb each addition before adding more. Stir often and cook until the risotto reaches an al dente texture and a creamy consistency, but not mushy. You may not need to use all the broth.

Return the mushroom mixture to the pot with the rice, stirring to combine.

Stir in the grated Parmesan and the thawed peas until well incorporated. Garnish with chopped fresh parsley before serving warm.