Shape the ground beef into 6 patties. Make them thinner than you would expect (½ inch).
Preheat a griddle to medium heat (325°).
Set aside 2 tbsp. butter. Spread the remaining butter on each slice of bread and place, butter-side-down, on the hot griddle. Top each slice of bread with a slice of American cheese. Allow to toast until the bread is golden and the cheese is melting. If it starts to get too brown, turn the griddle off and leave the bread while you cook the burgers.
Meanwhile, heat oil in a large cast iron skillet over high heat until hot, but not smoking. Add 2 tbsp. butter, which you had set aside. Add the burger patties to the pan. Season well with seasoned pepper. Cook over high heat for 1-2 minutes. Reduce heat to medium-high and cook for about 3 more minutes. Flip. Season again, and cook for about 4 more minutes for medium-well.
Place the burgers on the toasted bread. Top with remaining bread slices.