Add both mushrooms, onion, garlic and farro to a 6-quart slow cooker.
Whisk the broth, wine, tomato paste, soy sauce, thyme, salt and pepper and pour the mixture into the cooker.
Cover and cook on low until the farro is chewy-tender and the mushrooms are deeply browned-looking, 5 1/2 to 6 1/2 hours.
Uncover and cook on high for 10 minutes if more than a little broth remains at the bottom.
Stir in the butter until the liquid turns glossy and coats the farro.
Fold in the Parmesan and let the ragout stand for 5 minutes before spooning into bowls.