Slice the tomato and red onion; rinse and dry the green leaf lettuce; set out the dill pickle chips.
Divide the ground bison into 4 equal portions (about 6 ounces each). Gently form 3/4-inch-thick patties slightly wider than the buns and make a shallow dimple in the center of each.
Heat a cast-iron skillet over medium-high heat or preheat a grill to medium-high (450–500°F). Add the neutral oil to the skillet or oil the grill grates.
Season both sides of the patties evenly with the kosher salt and black pepper just before cooking.
Sear the patties until well browned on the first side, 3–4 minutes. Flip and cook 2–4 minutes more to 130–135°F for medium and juicy; cook to 160°F if you prefer fully done. In the last minute, top each patty with cheddar cheese and cover to melt, then transfer to a plate to rest 3 minutes.
Spread the unsalted butter on the cut sides of the burger buns and toast cut-side down in the skillet or on the cooler side of the grill until golden, 1–2 minutes.
Spread the mayonnaise on the toasted bottoms, add a rested patty to each, and layer the prepared toppings as desired. Cap with the top buns and press gently.