Heat the olive oil in a large pot over medium heat. Add the onion and cook for 3 minutes until it softens.
Stir in the garlic and cook for 1 minute more. Add the mushrooms and carrots. Cook for 5 minutes, stirring now and then, until the mushrooms release their liquid.
Pour in the farro, vegetable broth, thyme sprigs, salt, and black pepper. Bring the mixture to a boil.
Reduce the heat to low, cover the pot, and simmer for 30 minutes. Stir once or twice during cooking.
Remove the thyme sprigs before serving or storing.