Bring a large pot of water to a boil. Meanwhile, combine the chicken thighs, chicken broth, sliced ginger, scallion whites, 3/4 teaspoon salt and a few grinds pepper in a large saucepan. Cover and bring to a simmer over medium-high heat, about 8 minutes. Uncover and gently simmer until the chicken is cooked through, 5 to 10 more minutes.
While the chicken cooks, trim the snap peas and thinly slice. Peel the carrots and halve lengthwise, then thinly slice.
Add the soba noodles to the boiling water and cook as the label directs. Drain and rinse under warm water, then vigorously shake off the water. Divide among bowls.
Using tongs, remove the chicken to a plate. Discard the ginger and scallions from the broth and skim off any foam. Shred the chicken and return to the broth. Add the shiitakes and carrots, return to a simmer and cook until the vegetables are tender, about 3 minutes. Stir in the soy sauce and lemon juice; season with salt and pepper. Stir in the snap peas.
Ladle the soup over the noodles. Top with the scallion greens and grated ginger; sprinkle with shichimi togarashi.