Toss the onions with ¼ teaspoon kosher salt and arrange in a thin layer on a double layer of paper towel. Let the onions drain for 15 minutes. Gently pat dry.
While the onions are resting, divide the ground beef into two equal portions and shape them into patties about 1-inch thick. Season generously with salt and pepper.
Preheat the Blackstone or other outdoor griddle to medium.
Working quickly, place each burger patty on the hot part of the griddle and immediately smash it with a burger press until about ⅜-inch thick (1 cm). Cook until the underside of the burger begins to brown and develop a crust, 2 to 3 minutes.
Divide the onions equally between the burgers and pile them in an even layer on the raw side of the burger.
Flip the burgers so that the onions are now in direct contact with the griddle. Cook until the onions are softened and beginning to crisp, 1 to 2 minutes.
Top each burger with a slice of cheese and continue to cook until the cheese is melted, 1 to 2 minutes more. While the cheese is melting, place the burger bun halves cut side down on the cooler zones of the griddle to let them toast for 30 seconds.
Serve the burgers immediately, topped with sliced tomato and shredded lettuce, if desired, along with any other classic burger condiments.