Melt 2 tablespoons butter in a large frying pan over medium heat. Add the chicken pieces and cook for about 5 minutes, turning once, until golden brown on both sides. Season with black pepper.
Add the shallot halves and reduce heat to low, cooking for a further 6-7 minutes. Be sure to shake the pan and stir occasionally to prevent burning and sticking.
Transfer the chicken to a plate. Add the vinegar to the shallot halves in the pan and cook, stirring frequently, for about a minute, until the liquid is almost evaporated. Stir in the garlic, white wine, and chicken broth.
Return the chicken to the frying pan, along with any drippings that have accumulated on the plate. Cover and simmer for about 3 minutes, until the chicken is fully cooked.
Transfer the chicken pieces and shallots to a serving plate and cover to keep them warm. Increase the heat on the pan and boil the liquid until it's reduced by half, then remove the pan from the heat. Whisk in the remaining 2 tablespoons butter until the sauce is thickened, then add the parsely and pour the sauce over the chicken and shallots and serve.