Light your grill and set it up for two zone cooking, making the hot zone a medium high heat and the cooler zone medium heat. A griddle or cast iron pan also work well for this recipe.
While the grill is preheating, form the ground beef into four equal sized patties. Evenly coat the patties with olive oil, and season both sides liberally with kosher salt.
Cut the stem off the bananas. Then, slice them in half lengthwise, then again in half sideways so each banana is quartered. Remove from the peel and set aside.
Add the burgers to the grill, griddle, or cast iron pan over medium high. Sear on both sides and cook till they’re 5°F below your preferred serving temperature. Place two slices of smoked mozzarella cheese on each burger and place them on a plate. Tent with foil and allow carry over cooking to bring the burger up to desired temperature and melt the cheese.
While burgers are tented, evenly coat the cut side of the buns and the cut side of the bananas with remaining olive oil. Place them cut side down on the cooking service, directly over medium heat to toast the buns and caramelize the cut side of the bananas. The buns will toast quickly, as will the bananas. 1-2 minutes and these will be done.
Add the burger patty to the toasted bottom bun, top with chimichurri, two pieces of banana, and top bun. Serve with an extra side of chimichurri for dipping.