Season fillets with salt and pepper. Heat a frying pan (not nonstick) over medium-high heat; add oil, then fillets. Cook until fillets start to brown on one side, about 2 minutes. Flip fillets and reduce heat to medium. Add stock, bok choy leaves, mushrooms, soy sauce, and ginger. Cover and cook until fish is opaque in the center, about 3 minutes.
Divide fish, mushrooms, and bok choy between 4 deep, wide soup plates. Pour stock over fish and vegetables (depending on size of your plates, you may have extra stock). Garnish with green onion and jalapeño and drizzle with toasted sesame oil.