Bring a large pot of salted water to a boil. Add the soba noodles and cook until al dente, about 5 minutes. Drain the noodles, transfer them to a medium bowl and keep warm.
Meanwhile, in a small bowl, combine the soy sauce, lemon juice and 1 tablespoon of the ginger. Lightly brush the salmon fillets on both sides with a little of the soy-ginger mixture, and then dredge in the sesame seeds.
Heat the vegetable oil in a large nonstick skillet. Add the salmon to the pan, and cook over moderately high heat until the sesame seeds are golden and the fish is just cooked through, about 5 minutes per side.
Add the sesame oil, scallions, the remaining 1 tablespoon of ginger and the remaining soy-ginger mixture to the soba noodles and toss to combine. Transfer the soba noodles to serving bowls, top with the salmon fillets and serve.