Combine the water and kombu in a medium pot and let soak 10 minutes. Set over medium heat and warm until small bubbles form around the edges and steam rises, 8–10 minutes; do not boil.
Remove the kombu. Add the katsuobushi, bring just to a simmer, then turn off the heat and let steep 5 minutes. Strain through a fine-mesh sieve or cheesecloth into a clean pot to make dashi.
To the hot dashi, add the soy sauce, mirin, and sugar. Bring to a gentle simmer and cook 2–3 minutes until the sugar dissolves. Keep the broth hot over low heat.
Bring a large pot of water to a rolling boil. Add the dried soba noodles, stir to separate, and cook 4–6 minutes (or per package) until tender with a little bite. Drain and rinse under cold running water, gently rubbing the noodles to remove excess starch; drain well.
Divide the well-drained soba among 4 warm bowls. Ladle hot broth over the noodles (about 1.5 cups per bowl). Top with the sliced scallions and serve immediately.