Divide the ground beef (80/20) into 4 equal portions (about 6 oz each) and gently form 4 patties about 3/4 inch thick with a slight dimple in the center; keep chilled while you heat the pan.
Split the hamburger buns. In a large skillet over medium heat, melt the unsalted butter; toast the cut sides until golden, 1–2 minutes, then set the tops and bottoms aside and spread the mayonnaise on the cut sides.
Increase heat to medium-high, add the vegetable oil, and heat until shimmering. Season the patties on both sides with the kosher salt and black pepper, add to the skillet, and cook until well browned, 3–4 minutes. Flip, top each with American cheese, and cook 3–4 minutes more until the cheese melts and the patties reach 135–140°F for medium (or up to 160°F for well done); transfer to a plate to rest 3 minutes.
Reduce heat to medium. Crack the eggs into the skillet, using the remaining butter to keep them from sticking; cook sunny-side up until the whites are set and the yolks are still runny, 2–3 minutes (cover for 30–60 seconds if needed).
Assemble the burgers: set a patty on each bottom half, top with a fried egg, cap with the top halves, and serve immediately.