Form the meat into 4 equal square-shaped patties and season on both sides with salt and pepper. Heat a large cast iron pan over medium-high heat. Working 2 patties at a time, add to the pan and cook until a browned crust forms, about 3 minutes. Flip and continue to sear until cooked to medium, about an additional 2 minutes. Remove from the pan and set aside, covered on a plate. Repeat with the remaining patties.
Wipe out the pan and reduce heat to medium-low, add the vegetable oil and heat through then add 1 tablespoon of butter and allow to melt. Working in batches, add 2 slices of bread to the pan top with each with 2 slices of cheese, then follow with beef patties, a spoonful of caramelized onions and the remaining slices of Cheddar cheese. Spread the remaining 2 slices of bread with about 1 tablespoon of sriracha mayonnaise each.
Weigh down the sandwiches with either a heavy pan or brick wrapped in aluminum foil. Cook until golden brown, about 3 minutes. Flip the sandwiches and weight again, cooking until golden brown and the cheese is melty, about an additional 3 minutes. Remove from the pan.
Repeat with the remaining sandwiches and toppings. Open the sandwich and add a handful of arugula then close and slice in half.
Serve immediately with a side of pickles.