Preheat the grill to medium-high heat.
Heat the butter in a large heavy skillet over medium-high heat until hot. Stir in the yellow onion, 1 teaspoon salt, and 1/2 teaspoon pepper and cook, stirring occasionally until softened and golden, 20 to 25 minutes. Stir in the sugar, then pull the pan off the heat and carefully add the whiskey. Ignite the whiskey with a gas flame or a long kitchen match and cook until the flames die down. Continue to cook until the onions are glazed, about 2 minutes. Let cool, then season with salt to taste.
Break the bison up in a large bowl and season with 11/2 teaspoons salt and 1 teaspoon pepper, then divide and form into 4 (4-inch) patties.
Slice the white onion crosswise into 4 (1/3-inch-thick) slices. Skewer each onion slice to hold together, then rub with 1 tablespoon oil and sprinkle with 1/2 teaspoon salt and 1/2 teaspoon pepper. Grill onions, turning once, until softened, about 8 minutes.
Brush the grill grate with oil. Grill the burgers, turning once, until grill marks appear, 6 to 8 minutes total for medium-rare. Set aside to rest while you toast the buns.
Brush the cut sides of the rolls with the remaining 1 tablespoon oil and grill cut-side down until grill marks appear, 1 to 2 minutes.
Put a burger on the bottom portion of buns and top with lettuce leaves, whiskey-glazed onions, grilled onions, and bun tops. Serve.