Heat a 3 1/2-quart pan over medium-high heat. Add 2 tablespoons of the oil and heat for 30 seconds. Add the mushrooms to the pan and cook, stirring often, until golden brown and all of the water cooks out, 8 to 10 minutes. Remove the mushrooms to a plate with a slotted spoon. Add the remaining 2 tablespoons oil and the leeks. Reduce the heat to medium and cook the leeks, stirring often, until soft and fragrant, about 3 minutes. Add the garlic and salt and cook 1 more minute. Add the farro to the pan and stir to coat the grains in the oil and toast slightly, 2 to 3 minutes. Deglaze with the wine and stir. Cook the farro, stirring occasionally, until the wine evaporates almost completely, about 8 minutes. Add the broth in 1-cup increments, cooking until the liquid almost completely evaporates before adding the next increment; this will take about 45 minutes to complete. Remember to stir often to create a creamy texture.
When the farro is tender and absorbs almost all of the liquid, stir in the mushrooms, Parmigiano-Reggiano, Pecorino Romano and butter. Continue to stir until the butter is melted in and the mixture is creamy. Stir in the peas at the last minute and serve.
Recipe courtesy of Giada De Laurentiis