Place the dark chicken meat in a saucepan and cover with water; bring to a boil and cook until fully done, about 15–20 minutes depending on size.
Remove the chicken from the water and allow it to cool; reserve the broth if desired for another use.
Once cool, finely chop the chicken into small pieces and transfer to a mixing bowl.
Add the diced celery, salt, sugar, black pepper, sweet pickle relish, and mayonnaise to the bowl and stir until evenly combined.
Butter one side of each bread slice, then grill the bread butter-side down in a skillet or on a griddle until golden brown.
Divide the chicken salad among four sandwiches by placing portions between two grilled slices of bread and serve immediately.