Mushroom and Pea Farrotto

Name: Mushroom and Pea Farrotto
Ingredients: 1/4 cup extra-virgin olive oil; 10 ounces cremini or mixed mushrooms, sliced; 2 leeks, green parts and roots removed, halved lengthwise, then sliced into half moons and soaked in cold water to remove grit; 2 cloves garlic, smashed; 1/2 teaspoon kosher salt; 1 cup farro, rinsed; 1 cup dry white wine; 3 cups low-sodium chicken or vegetable broth; 1/3 cup freshly grated Parmigiano-Reggiano; 1/3 cup freshly grated Pecorino Romano; 4 tablespoons unsalted butter; 3/4 cup frozen peas, thawed