Season chicken with salt and pepper on both sides. Heat oil in medium soup pot over medium-high heat until just smoking. Brown chicken, skin side down, until golden, about 6 minutes; transfer to plate. The chicken will still be raw.

Add leeks, celery, and carrot to the fat left in the pot and cook over medium heat until softened, about 5 minutes. Stir in garlic and cook until fragrant, about 30 seconds. Stir in flour and cook for 1 minute. Slowly whisk in broth, scraping up any browned bits.

Add browned chicken, flesh side down, with any accumulated juices, and bay leaf. Cover and bring to a simmer. Cook on a low simmer for 20 to 22 minutes, or until the chicken is completely cooked. Transfer chicken to cutting board, let cool slightly, then shred into bite-size pieces using 2 forks, discarding skin and bones.

Meanwhile, cook pasta or noodles separately in a pot of boiling salted water, according to package directions. Drain and add to bottoms of bowls.

To serve soup, discard bay leaf. Off heat, stir in shredded chicken and let sit until heated through, about 2 minutes. Stir in parsley and season with salt and pepper to taste if needed. Pour hot soup over noodles in bowls and serve.