Set your oven to 350°F before you do anything else. A properly heated oven means the dip bakes evenly and the cheese melts right on schedule. While it warms up, you can get all your ingredients ready on the counter.
Stack the corned beef slices and cut them into small, bite-sized pieces. I like them tiny so you get a little bit of meat in every scoop. If your store does not carry Buddig, good deli corned beef works just fine.
Open the can and pour the sauerkraut into a strainer. Press down with a spoon to squeeze out as much liquid as you can. This step matters a lot, because wet kraut will turn your dip soggy.
Grate the Swiss and cheddar cheese with a box grater. Freshly shredded cheese melts smoother and tastes better than the pre-shredded kind. Measure out two full cups of each and set them aside.
Add the corned beef, drained sauerkraut, both cheeses, and mayonnaise to a large bowl. Stir well until every piece is coated and the mixture looks even. It will look thick and chunky, and that is exactly how it should be.
Scoop the mixture into a 9 x 13 inch baking pan. Use the back of a spoon to spread it into an even layer all the way to the edges. An even layer helps the dip heat through at the same speed.
Slide the pan into the preheated oven and bake for 20 to 25 minutes. You will know it is ready when the cheese is melted and the edges start to bubble. I peek at the 20-minute mark and pull it out when it looks golden.
Let the dip sit for a couple of minutes so it cools just enough to scoop. Bring it straight to the table with mini slices of rye bread or crunchy bagel chips. Watch how quickly the pan empties.