Slice about a 3-inch deep cut on the side of the chicken. Avoid cutting the chicken in half with the sharp knife.
Stir the hot sauce, lemon juice, and melted butter in a bowl. Include the celery salt and minced garlic into the mixture. Add a dash of pepper and salt to flavor. With a brush or your hand, season the whole chicken with the mixed sauce. Cover the entire interior and exterior of the meat. Reserve any remaining sauce for later.
Stuff the shredded mozzarella inside the chicken cuts. Wrap the ends of the chicken over the cut and tuck them inside securely. Push a toothpick through the meat to keep it together.
Heat the olive oil in a grill pan set over medium-high heat. Grill the chicken on every side for 5 minutes a side. Ensure that the chicken is cooked through and has no pink spots. For every flip of the chicken, brush the cooked sides with the remaining sauce for a more flavorful product.
Remove from the heat once cooked. Take out the toothpick. Serve in slices and enjoy while hot.