Evenly sprinkle ½ teaspoon kosher salt and ½ teaspoon fresh cracked black pepper on both sides of the steak. Set it aside.
Using a 10 to 12 inch non-stick skillet over medium-high heat, add the extra virgin olive oil and the tablespoon of butter.
Once the butter melts, add in the thinly sliced onion.
Sprinkle the onions with the remaining ½ teaspoon of kosher salt and cracked black pepper. Cook for 5 to 7 minutes, stirring frequently, or until golden. Remove the onions from the skillet, and set them aside.
Keep the skillet over medium-high heat, and add the seasoned steak. Cook for 3 to 4 minutes per side. Remove the steak from the skillet and allow it to rest for 5 minutes. Remove the skillet from the heat and wipe out the skillet.
Once the steak has rested, thinly slice the steak. Evenly divide the steak into 4.
Butter 1 side of each piece of bread.
Before returning the skillet to the heat, lay 4 of the slices of bread, butter side down.
Layer ⅓ cup of the shredded cheese, thinly sliced steak and cooked onions.
Top with ⅓ cup of the shredded cheese and the top slice of buttered bread.
Once all 4 sandwiches are assembled, heat the skillet to medium-high heat. Keep a close eye on the temperature, and adjust up or down to avoid burning the sandwich.
Once the bottom of the sandwich is golden, carefully flip the sandwich over. Remove the sandwich from the skillet and serve immediately.