Step 1

In a large bowl, stir together the flour and ½ teaspoon salt. Make a well in the center of the flour and add ⅓ cup water. Using a silicone spatula or wooden spoon, begin incorporating the flour by stirring at the center and gradually drawing in more, until a shaggy dough forms. Using the heel of your palm, knead the dough in the bowl, swiping along the sides to incorporate dry bits. Lightly flour the counter and turn the dough out onto it. Lightly flour your hands and knead until the dough is smooth, about 4 minutes. Form the dough into a ball, wrap it in plastic wrap and let rest at room temperature while you make the soup.

Step 2

In a large pot, combine 2½ quarts water, the chicken, onion, scallion whites, garlic, 1 tablespoon of the soy sauce, 1 teaspoon salt and ½ teaspoon pepper. Bring to a simmer over medium-high, then reduce to medium-low, cover partially and simmer until a skewer inserted in the thickest part of the chicken meets no resistance, about 25 minutes.

Step 3

Meanwhile, in a small bowl, stir together the scallion greens, remaining 2 tablespoons soy sauce, gochujang, sesame seeds, sesame oil and gochugaru (if using); set aside.

Step 4

When the chicken is done, transfer it to a large plate. To the broth, add the carrots and bring to a simmer over medium-high. Cook, uncovered and stirring occasionally, until the carrots are tender, about 10 minutes. Meanwhile, use 2 forks or your hands to shred the chicken into bite-size pieces; discard the skin and bones.

Step 5

Unwrap the dough. Working beside the pot, hold the dough in one hand. With your other hand, pull and stretch small sections of the dough, flattening it between your thumb and forefinger, then tear off a piece roughly 1½ inches long; drop the piece into the broth. Continue until you have used all the dough. Stir in the zucchini and shredded chicken. Return to a simmer and cook, stirring occasionally, until the noodles are tender and the zucchini is tender-crisp, about 3 minutes. Off heat, stir in the gochujang-sesame mixture, then taste and season with salt and pepper.