Add oil to a heavy-bottomed pan and heat to medium-high heat. While the pan is heating, separate the beef into 6 equal portions and shape into 1/2 inch high patties. Sprinkle patties on both sides with salt and pepper. Add the patties to the pan and let cook for 4 minutes. Flip the patties and let cook for another 4 minutes. Add the 1-2 cheese slices to the top of each patty and let melt for one minute.
Slice the buns in half and toast until lightly golden.
Take each bun bottom and top with burger patty, 2 tomato slices and 2 arugula leaves. Place the bun top on each patty. Serve on a platter with ketchup on the side.