Season chicken all over with 1 teaspoon salt and the pepper. Pat again to keep surfaces dry.
Heat olive oil and butter in a large Dutch oven over medium-high heat.
Sear chicken skin-side down until deep golden, 6–8 minutes. Flip and brown 3–4 minutes more. Transfer to a plate.
Add mushrooms and cook until browned and their liquid evaporates, about 5 minutes. Scoop out and reserve.
Soften onion, carrots, and celery in the pot over medium heat, 4–5 minutes. Stir in garlic for 30 seconds.
Sprinkle flour over vegetables and cook, stirring, 1 minute to remove raw taste.
Pour in white wine and scrape up browned bits. Simmer until slightly reduced, about 2 minutes.
Stir in broth, thyme, and remaining 1/2 teaspoon salt. Return chicken to the pot, skin-side up.
Cover and simmer gently until chicken is tender and cooked through, 20–25 minutes.
Uncover. Stir in cream, Dijon, lemon juice, browned mushrooms, and peas. Simmer 3–4 minutes to thicken.
Taste and adjust salt. Finish with parsley and serve hot.