Start by dividing fresh ground beef into 2-ounce balls, two balls per finished burger. Place the ball between two sheets of plastic wrap on the table.
Flatten the meat balls with the bottom of a stick skillet, pressing evenly and firmly.
The finished patties should be about 5 to 6 inches wide, and 1/4 to 3/16ths of an inch thick.
Place one slice of American cheese in the center of half of the patties. How you fit the cheese is up to you—it doesn't really matter, any way you slice it.
Place the remaining patties over the cheese-covered ones and pinch all the way around the edges several times until the seal is not visible.
Flip and Repeat
Flip the patties with a thin metal spatula and pinch around the perimeter again, to ensure against blowout.
Use a paper towel to apply a thin layer of oil to a heavy stainless steel or cast iron skillet and preheat it over medium-high heat until lightly smoking. Season your patty, add it to the skillet, and press down firmly with a spatula for 3-4 seconds to help contact between the patty and the skillet. Cook until well browned, about 2-3 minutes.
Flip the patty and continue to cook the second side until desired doneness is reached, about a minute for rare, two for medium rare, or three for medium. This is one case where I don't use a thermometer, as it would create an ooze-channel for the cheese.
Place the burger in a toasted bun with condiments of your choice (I go commando with this one, though onions and pickles are the traditional combo).