Preheat the oven to 425 degrees. Line a baking sheet with parchment paper.
Take the cabbage and slice 1-inch off the top. Next, slice the cabbage into 4 to 5, half inch thick steaks and place them on the baking sheet. Top with 3 tbsp olive oil, kosher salt, and pepper. Roast the cabbage for about 20 minutes.
While the cabbage is cooking make the Guinness caramelized onions. Melt 2 tbsp of butter in a skillet over medium heat. Next, add in the sliced onions, kosher salt, and pepper. Cook the onions for about 10 minutes and then pour in Guinness. Keep cooking until the onions are caramelized for about 10 more minutes. Remove pan from heat and set aside.
In a large mixing bowl, combine ground beef, kosher salt, pepper and 1 tbsp olive oil. Take the beef mixture and form into burger patties. Remove the cabbage from the oven and place the burger patty on top of the cabbage. Gently press creating a smash burger type topping. Place the baking sheet back into the oven to cook for about 8 minutes.
Remove the baking sheet again and top the burgers with sliced cheese. Place the sheet pan back into the oven to melt the cheese.
To serve: place the cabbage burgers on a platter. Top with Guinness onions and ketchup!