Combine the mayonnaise, grated onion, ketchup, relish, paprika, and salt in a small bowl until thoroughly blended to create the thousand island dressing. Lay out all slices of rye bread and place a slice of Swiss cheese on each. Evenly distribute the coleslaw over the cheese on half of the bread slices. On the remaining slices, layer the turkey. Generously spread the prepared dressing atop the turkey slices. Match each turkey-topped slice with a coleslaw-topped slice to form complete sandwiches.

Coat the exterior of each sandwich with softened butter. Set the sandwiches butter-side down into a skillet preheated over medium-low heat. Once in the pan, spread butter on the sides now facing up. Cover the pan loosely with aluminum foil and cook until the underside is a rich golden brown. Flip the sandwiches carefully, cover again, and grill the second side to an equal golden hue. Continue cooking until the cheese is fully melted and the interior is thoroughly warmed.