Combine all ingredients listed for the sauce. (Start with 1 teaspoon of horseradish, and add more to taste.) Let sit.
For the Ham Reuben Sandwiches
In a skillet, melt ½ teaspoon butter over medium heat. Add a sandwich size portion of the ham to the skillet. Move the ham slices around in the pan, turning frequently until warm.
Move the ham to the side of the skillet. Lightly butter one slice of the rye bread. Spread some of the horseradish sauce on the other side. Place the bread buttered side down into the skillet.
Place two slices of cheese on top of the bread. Top with the heated ham, more horseradish sauce and a portion of sauerkraut.
Top with another sauced and buttered piece of rye bread, buttered side UP.
When the bottom of the sandwich is brown and the cheese is slightly melted, carefully flip the sandwich with a spatula. Cook until the other side is browned.
*Repeat this process for more sandwiches.
Transfer to a cutting board or plate, and cut. Serve warm with a dill pickle and chips.