Season the flat iron steak with 3/4 teaspoon salt and the black pepper.
Heat the oil in a skillet over medium heat and cook the onion with the remaining salt for 10 minutes until soft and deep brown.
Move the onion to a bowl, raise the heat to medium-high and sear the steak for 4 minutes per side until it reaches 145°F.
Rest the steak for at least 3 minutes and slice it very thinly across the grain.
Butter one side of each bread slice, place four slices butter-side down, then layer each with cheddar, steak, onion, more cheddar and a second bread slice butter-side up.
Griddle the sandwiches over medium heat for 3 to 4 minutes per side until the bread is evenly brown and the cheese is melted.