Heat the olive oil in a large pot over medium heat. Add the onion, garlic, carrots, and celery. Cook for 5 minutes until the vegetables begin to soften.
Add the mushrooms and cook for another 4 minutes until they release some liquid.
Stir in the farro, vegetable broth, thyme, salt, and pepper. Bring the mixture to a boil, then reduce the heat to low. Cover and simmer for 25 minutes, stirring once or twice.
Add the baby spinach and lemon juice. Stir until the spinach wilts, then remove the pot from the heat.