In a large bowl, combine ground beef, finely chopped onion, salt, dried basil, and black pepper. Mix gently until just combined.
Divide mixture into 8 to 10 equal portions and shape each into a patty.
Preheat grill or broiler and cook patties about 6 inches from heat until desired doneness, roughly 5–7 minutes per side.
During the last minute of cooking, spoon a layer of pizza sauce over each patty and top with mushrooms, sautéed onions, green peppers, olives, and shredded cheese.
Close the grill or broiler to melt the cheese slightly and warm toppings.
Toast hamburger buns if desired, then assemble burgers by placing each patty on the bun and topping as preferred.
Serve immediately with your favorite sides and enjoy the pizza-inspired burger flavor.