Trim fat from meat. If necessary, cut brisket to fit into a 3-1/2- or 4-quart slow cooker. Place meat in cooker. Sprinkle with spices from packet. Spread sauerkraut over the brisket. Drizzle the 1/2 cup salad dressing over all.
Cover and cook on low-heat setting for 4 to 6 hours or on high-heat setting for 2 to 3 hours. Remove meat from cooker and place on cutting board. Thinly slice meat against the grain. Return sliced meat to the cooker and stir to combine with the cooking liquid.
Using a slotted spoon, spoon corned beef mixture onto 8 slices of the toasted bread. Top with cheese, additional salad dressing, and remaining bread. Makes 8 sandwiches.