For the onions, heat the olive oil in a large skillet over medium-high heat. Add in the onions, season with salt and pepper, and cook until tender and just turning golden, about ten to 12 minutes. Reduce the heat to medium-low and continue to cook for 15 minutes or until well browned. Add in the wine, turn the heat up to medium-high, and cook, stirring frequently for five minutes or until the wine has all evaporated. Remove the pan from the heat and set aside.
For the burgers, heat a cast iron griddle or skillet over medium-high heat. In a bowl, gently mix together the ground bison, minced shallot, and a bit of pepper just until combined. When the pan is hot, brush the bottom with the olive oil. Divide the bison mixture into four loose portions and sprinkle with salt. Place the burgers salt side down into the pan and press with the back of a spatula to flatten into approximately 10-12 centimeter rough patties. Season the burgers with some more salt and cook for about three to four minutes on the first side. Flip the burgers and brush the tops with half of the Dijon mustard and continue to cook for two to three more minutes or until fully cooked. Give the burgers one more flip, brush the other side with the remaining Dijon then transfer the burgers to a plate to rest for three to five minutes.
To serve, spread a little grainy Dijon mustard onto one side of the bun and top with the lettuce or arugula. Top with the burgers, a scattering of finely grated cheese, and a good spoonful of the onions.