Place the chicken along with the salt, pepper, onion, and celery stalks in a large stockpot. Cover with water and bring to a boil. Lower the heat and simmer until the chicken is cooked through. Remove chicken from pot, cool, and remove skin and bones, if using whole chicken. Reserve the liquid.
Dice the chicken and place in a large bowl. Add the chopped celery, eggs, seasoning salt, mayonnaise, lemon-pepper, 1/4 teaspoon pepper, and some reserved stock and mix well.
Butter 1 side each bread slice. Divide chicken mixture among 4 of the slices on the unbuttered side. Add whole pimentos, 2 to 3, and sliced deli Cheddar. Cover with the remaining 4 slices of bread, buttered side up.
Preheat an indoor grill/panini press. Place sandwiches on the grill/panini press and grill for a few minutes on each side.