Heat oil in a large pan over medium heat. Add onion and cook, while stirring occasionally, for about 6 to 8 minutes, or until the onion is soft and translucent. Add garlic, thyme, and salt. Cook for about 2 minutes more until garlic and thyme are fragrant. Add farro and stir until it is coated with oil, about 2 minutes.
The farro will take about 30 to 35 minutes in total to cook. Start by adding 1 cup of broth and the vinegar to the pan, while stirring occasionally, until almost all of the liquid has been absorbed. Continue to cook farro for about 15 minutes adding broth as needed, waiting until liquid is almost absorbed each time before adding more. Halfway through cooking, add the mushrooms to the pan, and continue to cook for about 10 minutes more, adding broth as needed. Stir occasionally. During the last 5 minutes of cooking, add the peas and cover. When farro is tender and peas are bright green, remove from heat and gently stir in the cheese. Season to taste with more salt and ground black pepper. Serve immediately.