Set a colander over a medium bowl. Add the cucumber, sprinkle with the ¼ teaspoon of salt and toss to combine. Let sit for 15 minutes.

Fill a medium bowl with ice water and set aside. Bring a large pot of water to a boil over high heat. Carefully lower in the eggs and cook for 6 minutes for jammy eggs to 12 minutes for hard-boiled. Using a slotted spoon or skimmer, transfer the eggs to the ice water and let cool completely.

Add the soba noodles to the same pot of boiling water and cook according to the package instructions. Drain, rinse under cold running water and set aside.

In a liquid measuring cup, whisk together the miso, tahini and ¼ cup of the water until smooth. Whisk in the remaining 1 cup water and the ginger until well combined.

Cut the dark-green parts of the scallions into 3-inch pieces and thinly slice them lengthwise. (Reserve the white and light green parts for another use.) Transfer the scallions to the ice water with the eggs to get them really crisp.

Use your hands to squeeze the cucumbers over the colander to get out any excess liquid. Stir the cucumber liquid into the miso broth. Taste the broth, and season with salt, as desired. In a medium bowl, stir together the cucumbers, garlic, vinegar, chili oil and sugar.

Peel the eggs and cut them in half.

Divide the soba noodles among bowls. Arrange the crushed ice around the edges of the bowls and evenly divide the broth among them. Top each portion with the cucumber and the seasoned liquid, 2 egg halves and some of the scallion. Serve, with more chili oil for drizzling.