Thinly slice the red onion and crumble the blue cheese; set both aside.
Divide the ground beef into 4 equal portions (about 6 oz each) and form 3/4-inch-thick patties slightly wider than the buns; press a shallow dimple in the center of each.
Preheat a cast-iron skillet over medium-high heat (or heat a grill to high). Lightly oil the skillet or grill grates with the neutral oil.
Spread the cut sides of the buns with the softened butter and toast in the skillet or on the grill, cut side down, until golden, 1–2 minutes; set aside.
Season both sides of the patties generously with the kosher salt and black pepper just before cooking. Cook the patties until well browned on the first side, 3–4 minutes.
Flip the patties and cook 2–3 minutes more, then top each with about 1.5 oz of crumbled blue cheese; cover and cook 1–2 minutes until the cheese softens and the burgers reach desired doneness (125–130°F for medium-rare, 135–145°F for medium, 160°F for well done). Transfer to a plate and rest 3 minutes.
Assemble the burgers: Place a lettuce leaf on each bottom bun, add a patty with melted blue cheese, top with sliced red onion, and cap with the top bun. Serve immediately.