Combine the ground beef with the ground corned beef. Divide into 4 equal portions and shape into patties. Brush both sides of the patties with oil and season generously with salt and pepper. Heat a grill, grill pan or skillet over medium-high heat. Grill the burgers for 5-6 minutes, flip and grill another 5 minutes. Top each burger with a piece of cheese and cook until the cheese starts to melt.
While the burgers are cooking, place the drained sauerkraut in a medium skillet over medium-high heat. Cook, stirring, until hot and all the liquid is gone.
Combine the mayonnaise, ketchup, and relish.
When the burgers are done, brush the cut sides of the hamburger buns with olive oil and grill until lightly toasted.
Spread the mayonnaise/ketchup mixture on both sides of the buns; place a burger on the bottom bun, top with the kraut and top bun.