Heat a 5 1/2-quart braising pan over medium-high heat. Add the oil to the pan and continue to heat. Using a paper towel, dry the chicken pieces well. Season all sides of the chicken with 1 1/2 teaspoons salt. Add the chicken to the hot pan skin-side down, and cook until the skin is golden brown and the chicken releases easily from the pan, about 4 minutes. Flip the chicken and cook the other side until nicely browned, about 4 minutes more. Transfer the chicken to a plate and set aside.
To the hot pan, add the carrots, celery and onion. Cook, stirring often with a wooden spoon, until the vegetables are fragrant and beginning to soften, about 3 minutes. Add the hazelnut liqueur and balsamic vinegar to the pan and cook, scraping up all the brown bits that may have stuck to the bottom of the pan. Simmer until the liquid is reduced by half, about 2 minutes. Stir in the chicken broth, olives, figs and oregano. Nestle the chicken pieces into the vegetables skin-side down and simmer, covered, for 30 minutes, flipping the chicken halfway through.
Transfer the braised chicken to a clean platter to rest. Simmer the sauce until reduced slightly, about 5 minutes. Pour the sauce over the chicken and sprinkle with the hazelnuts and parsley.