Cut corned beef brisket into 1-inch cubes and keep cold. In a food processor, pulse corned beef to consistency of coarse ground beef. Combine with ground beef and pepper, form six patties and refrigerate.

In a frying pan over medium heat sauté onions with 1-tablespoon butter for 5 minutes. Stir in Irish stout and simmer until pan is nearly dry and onions have a syrupy glaze, about 5 minutes; set aside and keep warm.

In a large cast iron skillet, heat canola oil over medium-high heat and cook patties in batches. Sear for about 3 minutes per side and transfer to a clean plate. Using tongs and a paper towel, wipe skillet of excess oil between batches, and when finished searing.

Spread each slice of rye on one side with butter. On the opposite side spread half of the slices with Dijon and half with cheese slices.

With the skillet on medium heat and working in batches, lay slices butter-side down making sure to have an equal number of Dijon topped and cheese topped slices. Top Dijon sides with onions and a beef patty. When rye is golden crispy, top patty with the cheese-topped slice of bread and serve.