Season chicken with salt and pepper. Heat 1 tablespoon each olive oil and butter in both a Dutch oven and medium skillet over medium-high heat. When foam subsides, add chicken pieces skin side down and cook until well browned, 4 to 5 minutes on each side. Spoon off all but 2 tablespoons fat from Dutch oven. Add chicken from skillet, arranging pieces in a single layer as much as possible. Add stock, partially cover, and bring to boil. Reduce heat to low and simmer until chicken is fully cooked, 20 to 25 minutes. Remove Dutch oven from heat, transfer chicken to bowl; cover with foil and set aside.

While chicken is simmering, drain off all but 1 tablespoon fat from the now-empty skillet. Add onion, mushrooms, and ¼ teaspoon salt; sauté over medium-high heat, stirring occasionally, until mushroom liquid evaporates and vegetables begin to brown, 6 to 8 minutes. Add wine; cook until almost all liquid evaporates, 2 to 3 minutes. Transfer to small bowl and set aside.

Heat remaining 2 tablespoons butter in now-empty skillet over medium heat until foaming. Add flour and whisk until golden in color, about 1 minute. Add half and half, whisking vigorously until smooth. Immediately whisk this mixture into hot chicken cooking liquid in Dutch oven. Bring to a boil over medium-high heat; reduce heat to medium-low and simmer, stirring frequently, until thickened to the consistency of heavy cream, 6 to 8 minutes. Stir in mushroom mixture, thyme, lemon juice, and nutmeg; season with salt and pepper to taste. Transfer chicken to pot and simmer until heated through, 2 to 3 minutes. Stir in 2 tablespoons parsley. Transfer chicken pieces to serving or individual plates, spoon sauce over and sprinkle with remaining parsley.