For the red chile russian dressing: Whisk together the mayonnaise, ketchup, horseradish, capers, chili powder, cornichons and chile de arbol in a bowl and season with salt and black pepper. Cover and refrigerate at least 30 minutes to allow the flavors to meld.
For the reuben sandwiches: Lay out 20 slices of the bread on a clean work surface, and spread each with a few teaspoons of the Red Chile Russian Dressing. Top each bread slice with 2 slices of the cheese and a dollop of the sauerkraut, followed by a few folded slices of the corned beef. Top each sandwich with another slice of bread.
Heat the butter in a large skillet over medium-high heat. Working in batches of 5, place the sandwiches in the pan and press down with a metal spatula. Cook until both sides are golden brown. Add more butter, if needed, before cooking the next batch. Keep the sandwiches warm in the oven on a large baking sheet.