Go light. A gentle simmer in sake infuses delicate fish with flavor.
Heat grapeseed oil in a large skillet over medium-high heat. Add mushrooms and sprinkle with 1/4 teaspoon of the salt. Cook, stirring often, until mushrooms release their liquid and begin to brown, about 6 minutes. Stir in scallions, carrots, and bok choy. Sprinkle with pepper and remaining 1/2 teaspoon salt. Cook, stirring often, until vegetables are tender and begin to turn golden, 3 to 5 minutes. Stir in ginger and garlic; cook, stirring constantly, for 1 minute.
Add sake, broth, tamari, and mirin; bring to a boil, stirring to loosen any browned bits from bottom of skillet. Reduce heat to low. Nestle fish in liquid among vegetables. (Fish will not be fully submerged.) Drizzle with sesame oil. Cover and cook until fish is cooked through and flakes easily with a fork, about 10 minutes. Remove fish and vegetables; divide among 4 plates. Increase heat to high and boil cooking liquid until thickened and reduced by half, about 6 minutes; drizzle over fish and vegetables.