Cook the soba noodles according to package directions, then drain and set aside.
Cut the chicken breast into bite-sized pieces and season with sea salt and black pepper.
In a small bowl, whisk together the tamari, grated fresh ginger, minced garlic, water, and arrowroot powder to create the stir-fry sauce.
Heat the sesame oil in a large skillet or wok over medium-high heat.
Add the chicken to the skillet and cook until golden brown and cooked through, about 5-7 minutes.
Remove the chicken from the pan and add the broccoli florets and sliced red bell pepper, sautéing until tender-crisp.
Return the chicken to the pan and pour the sauce over the mixture, stirring constantly until the sauce thickens and coats everything.
Serve the chicken and vegetable mixture over the cooked soba noodles and garnish with sesame seeds.