Preheat oven to 200°F.

Heat sesame oil in a large deep nonstick skillet over medium-high heat.

Meanwhile, combine the flour and ginger on a plate. Dip fish into flour mixture, turning to coat lightly. Cook in hot oil for 3 minutes per side or until fish is opaque. Transfer fish to four dinner plates with raised rims. Drizzle soy sauce over fish and place plates in the oven to keep warm while preparing noodles.

Add broth to skillet and, if desired, chili oil to the same skillet. Add noodles, cover, and bring to a boil over high heat. Uncover and stir the noodles to separate them. Reduce heat, cover, and continue to simmer for 3 minutes.

Meanwhile, cut bok choy crosswise through stems and leaves into 1/4-inch slices. Stir into noodle mixture and increase heat to high. Stir in mushrooms, if desired. Cook, uncovered, for about 2 minutes or until noodles are tender (bok choy will be crisp-tender).

Using a pot holder, remove plates from oven. Spoon noodle mixture alongside fish. Serve with additional soy sauce, if desired.