Combine the mayonnaise, chili sauce, pickle, lemon juice, and salt and pepper, to taste. Spread 1 side of each slice of bread with some of the dressing. Arrange half of the cheese on 4 of the slices. Divide the salmon, sauerkraut, and remaining cheese among the cheese-topped slices of bread and top each with 1 of the remaining bread slices, dressing-side down.
Heat the butter in a large skillet over medium-low heat; add the sandwiches and something heavy (such as a cast iron skillet, flat saucepan lid, or heatproof plate and a weight such as a food can) to firmly press the sandwiches down. Cook for 6 minutes a side or until golden and the cheese has melted. Cut the sandwiches in half and serve.