To make the Russian dressing:
Finely mince the shallot with the kosher salt either by hand or in a small food processor.
Combine the shallots and salt with the mayonnaise, chili sauce, horseradish, Worcestershire sauce, Tabasco, and paprika. Stir to blend together.
Makes about 1-¼ cups of dressing.
To make the Reuben sandwiches:
Spread a generous amount of the Russian dressing on one side of each slice of the rye bread.
Layer the corned beef on two slices of the bread.
Scatter the sauerkraut over the corned beef and top with slices of the cheese.
Top with the remaining pieces of bread, brushing the outside with melted butter.
Place the sandwiches, buttered side down onto a hot Panini machine or in a hot skillet. Brush the remaining side with melted butter.
Toast the sandwiches until the bread is lightly golden and the cheese has started to melt.