The Korean red chili paste levels up chicken noodle soup with the perfect amount of heat.
Halve and thinly slice 1 medium yellow onion. Cut 1/2 medium zucchini in half lengthwise, then slice crosswise into 1/2-inch-thick half-moons. Cut 4 medium scallions crosswise into three pieces each; finely chop a handful of the dark green parts (about 2 tablespoons) and reserve for garnish. Thinly slice 4 ounces fresh shiitake mushrooms.
Heat 1 tablespoon neutral oil in a large heavy-bottomed pot or Dutch oven over medium-high heat until shimmering. Pat 2 pounds bone-in, skin-on chicken thighs dry with paper towels, then add them to the pot, skin-side down. Season the chicken with 1 teaspoon kosher salt. Cook undisturbed until the skin is crisp and browned, and some of the fat is rendered, about 8 minutes. Flip the chicken and cook until the second side is golden-brown, 4 to 5 minutes. Transfer the chicken to a plate (it will not be cooked through at this point).
Add 1/4 cup gochujang to the pot and cook, stirring constantly, for 30 seconds. Add 1 cup of the low-sodium chicken broth and scrape up the browned gochujang from the bottom of the pot. Add the remaining 3 cups low-sodium chicken broth, onion, zucchini, remaining scallions, and shiitakes. Stir to combine and bring to a boil.
Return the chicken and any accumulated juices on the plate to the pot. Bring the heat down to medium-low. Cover and simmer, stirring occasionally, until the chicken is cooked through and the vegetables are tender, 15 to 17 minutes. Meanwhile, bring a large saucepan of water to a boil over medium-high heat.
Transfer the chicken with tongs to a clean cutting board. Use 2 forks to shred the meat into bite-size pieces; discard the skin and bones. Return the chicken to the soup and add 1 tablespoon fish sauce. Keep the soup at a low simmer until the noodles are ready.
Add 8 ounces dried Chinese wheat noodles to the boiling water and cook according to package directions until tender. Drain the noodles, then transfer them to serving bowls. Taste and season the soup with more fish sauce as needed. Ladle the soup, chicken, and vegetables over the noodles. Garnish with the reserved scallion greens.