Heat the olive oil over medium heat in a large soup pot. Add the diced onion, sliced leeks, carrots, celery, and 1 teaspoon kosher salt. Sauté, stirring occasionally, until the leeks are soft and the vegetables begin to caramelize, about 14-16 minutes. Taking your time here helps build sweetness and depth in the soup base.
Add the chopped garlic, turmeric, and poultry seasoning to the softened vegetables. Continue to sauté for another 2-3 minutes, stirring frequently, until the garlic and spices are very fragrant and just beginning to toast. This helps release essential oils for deeper flavor.
Pour in the chicken broth and coconut milk. Add the raw chicken thighs or breasts, ensuring they are submerged in the liquid. Scrape up any bits stuck to the bottom of the pot, partially cover, and bring to a bare simmer over low heat. Simmer gently for 15-20 minutes until the chicken is cooked through and the vegetables are tender. Be sure not to let it boil vigorously to prevent the coconut milk from splitting.
Remove the cooked chicken from the soup and let it cool slightly on a cutting board. Cut it into bite-sized pieces or shred it with two forks. Return the chicken to the pot along with the frozen peas (if using) and chopped fresh parsley. Bring the soup back to a gentle simmer and cook for about 5 more minutes until the peas are bright green and tender and the chicken is heated through. I find that using fresh parsley here really brightens the finished soup.
Taste the soup and season with additional kosher salt and black pepper as needed. Serve hot, optionally garnished with some extra fresh parsley. Enjoy!