Bring a medium pot of water to a boil for the soba; keep it at a gentle simmer.
Season chicken with salt and pepper. Heat neutral oil in a large pot over medium heat.
Sauté garlic and ginger until fragrant, about 30 seconds. Add mushrooms and carrots; cook 3 minutes.
Pour in chicken broth, 1 cup water, soy sauce, and mirin. Bring to a gentle simmer.
Slide in the sliced chicken. Simmer until just cooked through, 5–7 minutes, skimming if needed.
Stir in spinach and most of the green onions; cook until wilted, 1 minute. Add rice vinegar and red pepper flakes.
Cook soba in the simmering water per package, 4–5 minutes. Drain and rinse under cool water to remove starch.
Toss noodles with half the sesame oil. Taste the soup and adjust salt or soy. Finish with remaining sesame oil.
Divide noodles into bowls. Ladle hot soup over. Top with sesame seeds, extra scallions, and a splash of lime juice.