Chicken Noodle Soup

In a large, deep pot over medium-high heat, heat oil. Season chicken on all sides with 2 teaspoons salt, then add to pot skin side down. Cook, undisturbed, until golden brown on one side, about 5 minutes, then turn and cook until golden brown on the other side, about 5 minutes more.
Add stock, water, thyme, and bay leaf. Bring to a bare simmer over medium-high heat. Reduce heat to medium-low and simmer until an instant-read thermometer inserted into thickest part of breast registers 165°, 20 to 30 minutes.
Transfer chicken breasts to a cutting board. Continue to cook dark meat about 40 minutes more. Transfer dark meat to cutting board and let cool at least 10 minutes. Remove skin and bones and discard. Shred chicken into bite-sized pieces.
Meanwhile, discard thyme and bay leaf. Add onion, celery, and carrot and cook over medium-low heat, adjusting heat as needed to maintain a bare simmer and stirring occasionally, until just barely tender, about 5 minutes. Add egg noodles and cook, stirring occasionally, until just al dente, about 5 minutes more.
Stir in chicken and pepper; season with remaining 2 teaspoons salt, as needed.
Divide soup among bowls. Top with parsley and more pepper.
Make Ahead: Soup (without noodles) can be made 5 days ahead. Store in an airtight container and refrigerate. To serve, bring soup to a simmer in a large pot, add noodles, and simmer until noodles are al dente, about 5 minutes.