Whisk mayonnaise, 1 tablespoon sesame oil, miso, water, 1 teaspoon vinegar, and ginger together in small bowl; set sauce aside for serving.

Bring 4 quarts water to boil in large pot. Add farro and 1 teaspoon salt and cook until grains are tender with slight chew, 15 to 30 minutes. Drain farro well and return to now-empty pot. Stir in remaining 2 teaspoons sesame oil and remaining 1 teaspoon vinegar and cover to keep warm.

Meanwhile, cut tofu crosswise into 8 equal slabs, arrange over paper towel–lined baking sheet, and let drain for 20 minutes. Gently press dry with paper towels.

Spread cornstarch in shallow dish. Coat tofu thoroughly in cornstarch, pressing gently to adhere; transfer to plate. Heat 2 tablespoons canola oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add tofu and cook until crisp and browned, about 4 minutes per side. Transfer to paper towel–lined plate and tent with aluminum foil. Wipe skillet clean with paper towels.

Heat 1 tablespoon canola oil in now-empty skillet over medium-high heat until shimmering. Add mushrooms and ⅛ teaspoon salt and cook until beginning to brown, 5 to 8 minutes. Stir in sherry and cook, scraping up any browned bits, until skillet is nearly dry, about 1 minute; transfer to bowl.

Heat remaining 1 tablespoon canola oil in again-empty skillet over medium-high heat until shimmering. Add spinach, 1 handful at a time, and cook until just wilted, about 1 minute. Divide farro among individual serving bowls, then top with tofu, mushrooms, and spinach. Drizzle with miso-ginger sauce, sprinkle with scallions, and serve.