Bring the chicken or vegetable broth to a simmer in a small saucepan over low heat.

Heat olive oil and 1 teaspoon of butter in a small saute pan over medium heat.

Sauté the shallot until tender but not browned.

Add the sliced mushrooms and cook, stirring often, until they soften.

Add the dried thyme, salt, and freshly ground black pepper to taste.

In the risotto pan, add the remaining 1 teaspoon butter and stir in the rice.

Cook and stir for a minute or two until the grains are well coated.

Add the dry white vermouth and let most of the liquid evaporate.

Ladle in enough warm broth to barely cover the rice.

Adjust the heat so the liquid simmers briskly.

Once the broth is absorbed, replenish with the same amount of broth.

Continue to stir often, adding broth as needed, until the rice is cooked but still slightly firm in the center (al dente).

Stir in the fresh or frozen peas.

Stir in the grated Parmigiano-Reggiano cheese.

Serve immediately.