Cook the soba noodles in boiling water according to package directions, then drain and set aside.
In a medium bowl, toss the diced chicken breast with arrowroot powder, sea salt, and black pepper until every piece is lightly coated.
Heat the avocado oil in a large skillet or wok over medium-high heat until it begins to shimmer.
Add the chicken to the skillet in a single layer and sear until golden brown and cooked through, approximately 5 to 6 minutes.
Add the broccoli florets and sliced red bell pepper to the pan, stir-frying for 3 to 4 minutes until the vegetables are tender-crisp.
In a small ramekin, whisk together the tamari, freshly grated ginger, and minced garlic.
Pour the sauce over the chicken and vegetables, tossing constantly for 1 minute to allow the glaze to thicken.
Remove the skillet from the heat, fold in the cooked soba noodles and toasted sesame oil, and garnish with sesame seeds before serving.