Heat oil in a large Dutch oven over medium-high until shimmering. Sprinkle chicken evenly with 1 teaspoon salt.

Add chicken, skin side down, to Dutch oven; cook, undisturbed, until browned, 2 to 4 minutes per side, adjusting heat as needed to prevent overbrowning. Transfer chicken to a large plate, reserving 1 tablespoon drippings in Dutch oven; discard remaining drippings.

Add onion, carrots, and celery to Dutch oven; cook over medium-high, stirring occasionally, until just softened, 6 to 8 minutes. Add curry paste, garlic, ginger, and brown sugar; cook, stirring constantly, until fragrant, 15 to 20 seconds.

Stir in broth and turmeric, scraping up any browned bits on bottom of Dutch oven; bring to a boil over medium-high, then reduce to a simmer over medium-low. Return chicken and any accumulated juices on plate to Dutch oven. Cover and simmer, undisturbed and adjusting heat as needed to maintain a simmer, until an instant-read thermometer inserted into thickest part of chicken registers 165°F, 15 to 20 minutes. Transfer chicken to a clean large plate.

Bring broth mixture to a boil over medium-high. Add noodles, and remaining ½ teaspoon salt; partially cover, and cook, stirring occasionally, until noodles are al dente, about 5 minutes.

While noodles cook, shred chicken; discard skin and bones.

Uncover Dutch oven and reduce heat to maintain a simmer. Add shredded chicken along with coconut cream, lime juice, and fish sauce, stirring until well combined. Add more lime juice to taste. Divide evenly among 6 bowls. Garnish with cilantro.