In a small bowl, stir together 2 tablespoons unsalted butter (room temperature), 1 minced garlic clove, and 1 tablespoon finely chopped fresh parsley until evenly combined.
Slice the 4 hamburger buns horizontally. Spread the garlic‑parsley butter evenly on the cut sides of all bun halves.
Heat a skillet over medium heat. Place the buns buttered side down in the skillet and toast until golden brown and crisp, about 2–3 minutes. Remove the toasted buns and set aside, toasted side up.
In a large mixing bowl, combine 1 pound lean ground beef, ½ medium yellow onion (finely chopped), 1 teaspoon garlic powder, 1 teaspoon Italian seasoning, 1 teaspoon sea salt, and ½ teaspoon freshly ground black pepper. Mix gently until ingredients are evenly distributed.
Divide the beef mixture into four equal portions and shape each into a patty slightly larger than the hamburger buns (they will shrink as they cook). If desired, press a small shallow dimple in the center of each patty to help them cook evenly.
Warm 2 tablespoons olive oil in a clean skillet over medium‑high heat until shimmering. Add the patties (do not overcrowd the pan) and cook about 4–5 minutes on the first side.
Flip the patties and cook the second side about 4–5 minutes more, until nearly at your desired doneness. During the last minute of cooking, spoon the pizza sauce evenly over the four patties (use the full 1 cup divided among them), sprinkle the 1 tablespoon chopped fresh basil over the sauce, top each patty with 1 slice mozzarella cheese, and place 3 pepperoni slices on each patty (12 pepperoni slices total).
Reduce the heat to low, cover the skillet briefly, or leave the pan on the heat just long enough for the cheese to melt and the pepperoni to warm, about 30–60 seconds.
Transfer each finished patty onto the bottom half of a toasted bun. Close with the top bun halves and serve the pizza burgers immediately while hot and the cheese is gooey.