In a large mixing bowl, combine finely chopped corned beef, shredded Swiss cheese, well-drained sauerkraut, and softened cream cheese. Add in ¼ cup of breadcrumbs and one beaten egg, mixing thoroughly until the ingredients are well-blended.

Scoop a portion of the filling mixture and roll it into 1 to 1.5-inch balls, placing on a tray lined with parchment paper. Chill in the freezer for 30 minutes.

Set up your breading station with flour, beaten egg, and remaining breadcrumbs. Roll each chilled ball first in flour, then dip in egg, and finally coat with breadcrumbs.

Heat oil in a deep fryer or skillet to 350°F, adding coated balls in batches. Fry for 3-4 minutes or until golden brown.

In a mixing bowl, whisk together mayonnaise, ketchup, relish, hot sauce, and smoked paprika for the dipping sauce. Adjust seasoning as needed.

Serve the Reuben Balls hot alongside the Thousand Island dipping sauce.