In a Dutch oven or 5-quart saucepan, heat oil over medium. Add ginger and garlic; cook 1 minute until fragrant. Add bok choy stalks and celery; season with salt and pepper and cook, stirring frequently, until crisp-tender, 5 to 7 minutes. Add scallions and bok choy greens and cook until wilted, about 3 minutes more.
Heat broiler with rack 4 inches from heat. In a small bowl stir together hoisin, ketchup, mustard, and 1/2 teaspoon lemon juice. Place fish on a broiler pan or broiler-proof rimmed baking sheet. Season with salt and pepper. Brush tops with hoisin mixture. Broil until fish is glazed and opaque throughout, 5 to 7 minutes.
Add remaining lemon juice to greens. Spoon vegetables on four plates and top with fish fillets. Serve immediately.