In a medium saucepan, heat 1 tablespoon 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 to 40 minutes. Season with salt and cover to keep warm.
Meanwhile, heat oven to 450 degrees. On a rimmed baking sheet, toss mushrooms with remaining 2 tablespoons oil and season with salt and red-pepper flakes. Roast, stirring once, until crisp and golden, 20 to 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.
Change It Up
Substitute wilted mustard greens, pine nuts, and currants for the mushrooms and spinach; stir in roasted cauliflower.