Special equipment: Chimney starter, all-natural lump charcoal and kettle grill
Pass the meat through the coarsest die of a food grinder. Place the ground meat into a bowl and mix together to evenly distribute the different types of meat. Place in the refrigerator uncovered, for 2 hours.
After 2 hours, fill a large chimney starter with charcoal and light. Once the coals are ashy white, transfer to a kettle grill and spread evenly. Set grate over coals.
While the coals are heating, shape the meat into 4 patties that are 6 ounces in weight and 3/4-inch thick. Sprinkle each patty on both sides with the salt. Place on the grill and do not move for 2 minutes. Flip the burgers every 2 minutes until they reach an internal temperature of 135 degrees on an instant-read thermometer. Watch for hot spots and move the burgers if flare-ups occur. Resist the urge to smash down on the patties. Remove the burgers from the heat to a plate and prepare the buns by placing them on the grill, cut side down for 1 to 2 minutes. Remove and place them cut side up on a platter. Divide the cheese evenly between the top buns, set the meat patty on top of the cheese. Coat the cut side of each bottom bun with a little mayonnaise and set atop the meat. Allow each burger to sit upside down for a minute before flipping and consuming.