Preheat the oven to 350°. Season both sides of the chicken with 1½ teaspoons of salt and the black pepper. In a large, deep, oven-safe skillet or pot set over medium-high heat, add the olive oil. Once the oil is shimmering, add the chicken thighs, skin-side down, and reduce the heat to medium. Cook until the skin is golden brown, 8 to 10 minutes. Use a metal spatula to turn the thighs over and cook until the surface of the meat pales, about 1 minute longer. Transfer the chicken skin-side up to a plate.
To the skillet, add the celery, onion, garlic, raisins, bay leaf, rosemary, thyme and remaining ¼ teaspoon salt. Cook, stirring often until the onions are soft, about 5 minutes. Increase the heat to medium-high and stir in the mustard, then pour in the wine. Cook until the wine is slightly reduced, 3 to 5 minutes, then add the broth and nestle the chicken thighs skin-side up into the skillet.
Cover the pan and braise the chicken in the oven for 25 minutes. Uncover and cook 20 minutes more. Transfer the chicken to a platter and set aside.
Set the skillet over medium-high heat and add the cream. Cook, stirring occasionally, until the sauce thickens, 3 to 4 minutes. Turn off the heat, remove the bay leaf, rosemary and thyme, and stir in the chopped celery leaves and parsley leaves. Spoon the sauce over the chicken and serve.