Heat the broiler. On a foil-lined baking sheet, arrange the chicken skin-side-down and, using a paring knife, lightly score the meat every 1⁄4 inch and season lightly with salt. Broil the chicken thighs, basting every 4 minutes with 1⁄4 cup of the sauce base and flipping halfway, until cooked through and golden brown, about 16 minutes. Transfer the chicken to a cutting board, let rest for 10 minutes, then slice each thigh into 3 thick slices.
In a large saucepan, combine the remaining 3⁄4 cup sauce base with the dashi. Using a Microplane set over a fine sieve set in a bowl, grate the 6-inch piece of ginger into the sieve, pressing on the solids to drain as much juice as possible into the bowl. Pour 2 tablespoons of the ginger juice into the saucepan and discard the rest or save for another use. Bring the soup to a boil over medium heat and keep warm.
In a large pot of boiling water, cook the soba noodles until al dente, about 3 minutes. Using tongs, lift the noodles from the water and transfer to a colander and rinse under cold running water until the water runs clear. Drain the noodles again and divide among 4 large serving bowls. Add the spinach to the boiling water and cook until just wilted, about 1 minute. Drain the spinach, pressing to remove as much water as possible, and divide among each serving of noodles.
Ladle the warm soup over the spinach and noodles in each bowl and top each with 3 chicken slices. Place 1 egg half and one-quarter of the chives in each bowl and then garnish with one-quarter each of the julienned ginger and both sesame seeds. Garnish each bowl with schichimi togarashi and serve immediately.