Heat a large pot of salted water to boiling. Add soba noodles and cook until al dente, about 5 to 7 minutes. Drain, rinse under cold water, drain again. Toss with 1 teaspoon oil and place in the refrigerator to cool while you prepare the rest of the dish.
Place chicken breast, ginger slices, and green onion pieces in a small saucepan and add enough water to cover chicken plus about 1/2-inch. Bring water to a boil over high heat, reduce heat to a simmer and cook, turning once, for about 15 minutes or until chicken is cooked through.
Remove from water, let cool slightly and use a fork to shred chicken. Set aside.
Prepare sauce by mixing all sauce ingredients together in a blender or food processor. Alternately, whisk sauce ingredients together by hand until emulsified.
Place cooked chilled or room temperature noodles in a large bowl with the shredded chicken and veggies. Pour sauce over everything and toss well to mix. Divide between two plates. Sprinkle chopped peanuts over the top; serve immediately.