Slice the onion and place in a dry skillet over medium-high heat. Toss around occasionally. Once they begin to stick and darken slightly, pour in about 1/2 of the beer. It will bubble up and sizzle. Reduce the heat slightly to about medium. Push onions under the beer and toss around periodically to evenly heat the onions. When they start to stick again, add more beer. If the onions are not completely caramelized once all of the beer is added, you can add a bit of water to help them wilt and soften slightly. Stir in the dijon mustard.
Taste. Is it too bitter or sweet? Stir in maple syrup to your preferred taste, while still keeping the onions with a hint of the bitterness.
Meanwhile, gingerly shape the ground bison into patties. Avoid pressing, smashing and man-handling the ground bison too much. Generously salt each side of the patty and add a pinch of black pepper too. The rest of the salt can be tossed in with the caramelized onions.
Bring a grill pan or outdoor grill up to smoking hot heat. Once it is hot, reduce the heat slightly to medium-high and then sear the burgers on the first side for about 6-8 minutes, uncovered (or if you want them well done, cover it). When dark grill marks appear, flip over the patties and add the cheddar cheese slices to the cooked side. Sear for another 4-5 minutes and then cover for the final couple of minutes to help the cheese melt a bit further.
Split and toast the buns on the grill or in a toaster/oven. Smoother each bun side with mayo, mustard or your preferred toppings such as ketchup, pickles, etc.
Place the cooked burgers in the bun and top with the caramelized onions and baby greens. Smash together and enjoy!