Heat grill to high heat on one side and medium heat on the other.
Divide the ground beef in half and form two thick burgers (or for smaller portions, form into 4 burgers).
Mix the two coarsely ground peppers together and place on a large plate. Press each burger into the pepper mixture, coating the ground beef with a thick layer of peppercorns. Flip and coat the other side with a thick layer of peppercorns.
Place a large cast iron skillet on the hottest part of the grill and allow it to get as hot as possible.
Add butter and oil and when that melts and starts to smoke a bit, place peppered burgers in the center of the pan.
After two minutes, turn each burger. After two more minutes, transfer the burgers out of the pan and place directly on the grates onto the cooler side of the grill. (If you formed four smaller burgers, they will cook in less time)
Place cut mushrooms and shallots into the hot pan from which you just removed the burgers, and cook and stir while burgers are gently cooking on the cooler side of the grill.
Turn the burgers once after about a minute and a half and cook for the same amount of time on the other side. (Again, less if you are cooking smaller burgers)
After the shallots and mushrooms have started to brown, pull the pan back and add the cognac being careful if it flares up.
Allow the shallot and mushroom mixture to cook until the cognac has evaporated, then add the cream. Stir and add burgers back into shallot and mushroom mixture and turn once to coat both sides in the sauce.
Top each burger with the slices of Wisconsin Pepato cheese and close the grill cover until the cheese has melted.
To serve, spoon the shallot and mushroom mixture from the pan onto the bottom half of each roll. Place the burgers on top of the shallot mushroom mixture, then top with a slice of fresh tomato and arugula.
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