Preheat a grill, cast iron skillet or flat top over medium-high heat.
Add the elk to a bowl with the salt and pepper.
Shape the ground meat into 4 evenly sized patties.
Add a quarter of the grated butter to the center of each burger and fold it in half. Shape into a disc again, making sure all of the butter is now in the inside of the burger. Repeat for each burger.
Place the burgers on the grill and grill for about 4 minutes per side or until they’re medium temperature (140-145F). During the last 2 minutes on the second side, add a slice of cheese to each burger.
Serve with any desired toppings on a bun.
You might want to grate your butter ahead of time and store in the fridge right until you need it so that it stays firm.