Heat 1 tablespoon butter in a medium saucepan over medium-high heat. Add the scallion whites and cook until light brown, about 3 minutes. Add the sauerkraut and apple and season with salt and pepper. Cook until the sauerkraut is lightly browned and dry, about 10 minutes. Add half of the scallion greens and cook for 1 minute.
Preheat the broiler. Mix the mayonnaise, chili sauce, the remaining scallion greens, the capers and their liquid, and the dill pickling liquid.
Spread about 1/2 teaspoon butter on one side of each bread slice. Place 4 slices, buttered-side down, on a foil-lined baking sheet. Spread each with some of the mayonnaise mixture, then layer the turkey, dill pickles and sauerkraut mixture on top. Sprinkle with the cheese and broil until the cheese melts, about 2 minutes.
Spread more of the mayonnaise mixture on the unbuttered sides of the other 4 bread slices; place buttered-side up on the sandwiches. Return to the broiler until golden, turning once, about 2 minutes.
Per serving: Calories 625; Fat 33 g (Saturated 12 g); Cholesterol 102 mg; Sodium 2,381 mg; Carbohydrate 46 g; Fiber 6 g; Protein 39 g