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.