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Cook chicken, skin-side down, in a large heavy pot over medium-high heat until the skin is golden brown and the fat has rendered, about 6 minutes. Transfer the chicken to a plate; remove and discard the skin.
Add mushrooms and onion to the drippings in the pot; cook over medium-high heat, stirring occasionally, until the mushrooms are browned and the onion is softened, about 7 minutes. Add carrots, garlic and thyme; cook, stirring constantly, until fragrant, about 1 minute. Add flour; cook, stirring constantly, until toasted, about 1 minute. Add milk, broth and salt; stir to combine. Nestle the chicken into the mixture. Bring to a boil over high heat; reduce heat to maintain a simmer. Cook until an instant-read thermometer inserted into the thickest portion of the chicken registers 165 degrees F, 20 to 25 minutes. (Do not remove from heat.)
Transfer the chicken to a cutting board; let rest for 5 minutes. Shred the meat into bite-size pieces; discard bones. Stir the shredded chicken and peas into the soup. Serve hot.