Prepare the burger sauce by mixing the mayo, steak sauce, Worcestershire sauce, Dijon mustard, and black pepper in a small bowl, then set aside.
Divide the ground beef into four separate balls, ¼ pound each. Firmly flatten the ground beef into thin patties with a burger press or by using a flat plate lined with parchment paper to prevent the meat from sticking. The patty should be wider than the buns to accommodate shrinking as they cook.
Place two slices of cheese on top of one patty, leaving ½" of space around the edge. Place a second patty on top of the cheese and pinch the edges shut to seal the patties together. Season each side with salt and black pepper. Repeat this process with the remaining two patties.*
Melt one tablespoon of butter in a large skillet set over medium heat and add the sliced onions. Saute the onions until tender, then set aside.
Place the cheese-stuffed burger patties in the skillet and cover with a lid. Cook until the patties are well seared on the bottom, then flip and continue until they are fully cooked, approximately 8 minutes, or when the internal temperature reaches 155 degrees. Due to their size, it may be necessary to flip multiple times to prevent burning or reduce the heat to medium-low.
While the burgers cook, butter the interior of the hamburger buns and toast in a separate skillet until golden brown.
Assemble the burgers by placing the cheese-stuffed patty on the toasted bottom bun, and top with sliced pickles, sauteed onions, burger sauce, and the top bun.
Let the burger rest briefly, then serve.
Notes
*Not fully sealing the two patties together will result in melted cheese oozing out as the burger cooks. If you can still see the line where the two patties connect, it isn't pinched together firmly enough.