Divide the beef into 2 patties, each about ¾-inch (2 cm) thick and 3 ½ inches (9 cm) in diameter. Transfer the patties to a plate or a baking sheet lined with parchment paper and refrigerate for 30 minutes.
Remove the patties from the fridge and season on both sides with salt and pepper.
Heat a large cast-iron skillet over medium-high heat. Add the oil to the pan and swirl to evenly coat. Place the patties in the skillet and cook, undisturbed, on the first side for 5 minutes, until a brown crust forms.
Flip the patties, top each with a slice of cheese (optional), and cook, undisturbed, until the patties are browned on the other side, the cheese is melted, and the meat is cooked through, about 5 minutes more for medium-well doneness.
Remove the patties from the pan and transfer to a clean cutting board. Let rest for about 5 minutes.
Place the burger patties on buns and serve with your favorite toppings and sauces.