Spread butter onto the inside of the top and bottom bun slice and lightly toast, butter side down, in 12" skillet on medium heat until the cut side of the bun develops a crispy, toasted, exterior. Turn off the heat once done.
Divide 1 lb of beef into three equal sections and form into round patties, being careful not to over work the meat, yet pressing firmly enough for the meat to hold together. Press an indentation into the center of each patty to prevent it from puffing excessively during cooking.
Season each side of the patty liberally with salt and pepper.
Add enough butter to coat the bottom of a cast-iron skillet heated to medium high and gently place in the patties once it has melted. Work in batches if necessary.
Once in the pan, do not move the patties, or press/squish the meat. Moving the patties unnecessarily will prevent a good sear from forming and pressing the meat is a deadly sin that causes delicious juices to escape.
After ~2 minutes of cooking, flip each patty and add a slice of cheese to the top of the patty while the burger finishes cooking.
Remove from the skillet while burgers are rare, or medium rare (~125 degrees). Overcooking Wagyu beef will cook out all of the flavorful goodness that makes Wagyu a premium product.
Place patty on toasted bun slathered with roasted garlic aioli, and topped with caramelized onions.