In a medium saucepan, heat 1 tbs oil over medium heat. Add shallot and cook, stirring, until tender, about 5 minutes. Add farro, stirring until toasted, 1 minute. Add wine and reduce by half. Add stock, bring to a boil, then reduce heat and simmer, stirring occasionally, until farro is tender and creamy, 35-40 minutes. Season with salt and cover to keep warm.
Meanwhile, heat oven to 450°F On a rimmed baking sheet, toss mushrooms with remaining 2 tbs oil and season with salt and pepper flakes. Roast, stirring once, until crisp and golden, 20-22 minutes.
Warm farro over medium heat and add spinach, stirring until wilted, about 1 minute. Stir in mushrooms and parmesan. Top with additional parmesan.