Lay the rye bread slices on a clean work surface and spread 1 side of each slice with some of the Russian dressing. Layer 1/2 of the slices with some of the corned beef. Divide the sauerkraut evenly over the meat, then top the sauerkraut with the slices of cheese. Place the remaining bread slices on top, dressing sides down, and set aside.
Lay out the rye bread on a clean work surface. Assemble the sandwich by adding Russian dressing to each slice of bread and then layer 1/2 of the slices with the corned beef. Split the sauerkraut and layer on top of the meat. Add 2 slices of cheese on top of the sauerkraut and then top with bread slice.
Heat a large frying pan or griddle on stove top on medium. Gently butter one side of each sandwich, and then place that side of the bread down in the hot pan. Cook until slightly browned, bout 4 minutes.
Spread the rest of the butter on the unbuttered top side of the sandwich. Gently flip the sandwiches to cook on the second side until golden brown, crispy, and the cheese is melted, 3 to 4 minutes longer.
Slice sandwich in half and serve hot!