In a large skillet over medium heat, add the ground beef and diced onion. Cook until the meat is browned and the onions are translucent, about 6-8 minutes. Drain excess fat if necessary.
Stir in the ketchup, yellow mustard, mayonnaise, white vinegar, dill pickle relish, garlic powder, onion powder, salt, and black pepper. Mix well until everything is combined.
Reduce the heat to low and let the mixture simmer for about 10 minutes, stirring occasionally, until heated through and slightly thickened.
During the last minute of cooking, place a slice of American cheese on top of the meat mixture to melt.
Slice the hamburger buns and toast them lightly if desired.
To assemble, spoon the Big Mac Sloppy Joe mixture generously onto the bottom half of each bun. Top with shredded lettuce, diced tomatoes, and sliced pickles. Place the top half of the bun on top.
Serve immediately with extra toppings on the side if desired.