Heat the oven to 350 degrees F.
Using a sharp knife, cut a pocket into each chicken breast. Stuff 1/4 of the blue cheese into each breast. Secure the openings with toothpicks and season with salt and pepper. Heat the oil in a nonstick, oven-proof skillet over medium heat and sear the chicken on both sides until browned, about 3 to 4 minutes per side. Put the chicken into the oven until it is cooked through, about 8 to 10 minutes.
In a small pot, add the butter, hot sauce, and white vinegar and cook over medium heat until bubbling. Pour half the sauce over the hot chicken and cook over medium heat for 1 minute. Remove the toothpicks and serve with extra sauce garnished with sliced celery.