Combine all the ingredients in a medium saucepan and bring to a simmer. Cook over low heat for 10 minutes. Strain the broth and return it to the saucepan. Boil over high heat until reduced to 2 cups, about 15 minutes. Season with salt and pepper and remove from the heat.
In a saucepan of boiling water, cook the somen until tender but still slightly chewy, about 3 minutes. Drain and rinse under cold water; drain.
Heat the oil in a wok. Add the shiitakes, snow peas, cabbage, bean sprouts, daikon, onion and carrot and season lightly with salt and pepper. Stir- fry over high heat until crisp-tender, about 3 minutes. Add the sesame oil and soy sauce; remove from the heat.
Season the fillets with salt and pepper. Heat the oil in a large skillet until shimmering. Add the fish, skin-side down, and cook over moderately high heat until the skin is crisp, about 4 minutes. Turn the fillets and cook until just opaque throughout, about 2 minutes longer.
Bring the broth to a boil. Stir-fry the vegetables over high heat to rewarm them. Put the noodles in 4 shallow soup bowls and pour the boiling broth over them. Spoon the vegetables over the noodles and top with the fish, skin-side up. Sprinkle with the chives, basil, cilantro and sesame seeds and serve.