Cook the Onions: Place a large skillet over medium heat. Add oil, the chopped onions and salt. Sauté for 5 to 8 minutes stirring regularly, adding a tablespoonful of cold water when required to prevent sticking. Once the onions are caramelized brown, remove from heat. Set aside.
Make the Animal Style Sauce: In a medium bowl, mix the ketchup, mayonnaise, relish and vinegar. Set aside.
Prepare the Toppings: Prepare the pickles, lettuce, tomato slices and set aside.
Make the Beef Patties: On a work surface, shape the ground beef into 8 patties about a ½ inch thick using your hands. Season with salt and pepper and set aside.
Prepare the Buns: Brush the buns lightly with oil and place on a skillet for 2 to 3 minutes. Spread a tablespoon worth of the prepared animal style sauce onto each bun followed by the prepared toppings to your liking. Set aside.
Cook the Burgers: Place a large skillet over medium-high heat. When hot add the beef patties. Slather a teaspoon of mustard onto each patty and cook or 3 to 4 minutes. Flip the patties and cook 3 minutes more, or until the beef reaches an internal temperature of 160°F. Add cheese on top of the burger during the last 1 minute.
Assembly: Spread animal style sauce on top of a bun. Then place 1 cooked patties onto a bun. Add with a spoonful of caramelized onions. Then place a second patty on top (this is the double double). Finally, add other toppings followed by the top bun. Repeat for the remaining patties to make 4 double-double burgers in total.