Slice the onion into a medium size bowl, add 1 teaspoon salt and toss to coat. Transfer the onion to a colander and let rest for 30 minutes – tossing occasionally. Place the onion in a clean dish towel (I used Bounty toweling and it worked out fine) and squeeze the onion dry. (This is a good time to butter and toast the buns)
Divide the onion into 4 equal portions and place them in a mound on a baking sheet leaving enough distance between them to form a burger. Divide the burger into 4 equal portions and form a ball out of each one like you are making a meatball. Place each meatball an onion mound; using your hands press the burger down over the onion to form a burger.
In a large nonstick or cast iron skillet, melt the butter with the oil over medium-high heat. Cook the burgers onion side down until nicely browned and onion is beginning to crisp on the edges. Flip the burgers, increase the heat to high, and continue cooking to desired doneness.
Place a slice of cheese on the bottom side of each bun, followed with the burger and any extra condiments. Serve hot. (You can also place the cheese on the burger while still in the pan, cover, and cook until cheese is melted)