Quarter the beef into even portions. Form into burgers, indenting the middle slightly with your thumb. Season with salt and pepper, and brush lightly with vegetable oil. Cook on a hot grill until charred on the outside and pink on the inside, about 4 minutes per side. Two minutes before removing the burgers, top the burgers with the mozzarella cheese.
Place the toasted muffins on a serving platter. Place the tomato slices on the bottom, and top each with cooked patties. Spread a tablespoon or two of the basil sauce on each bun top and sprinkle with a little Parmesan cheese. Serve extra basil sauce on the side for dipping.
Blend the green onion, basil, garlic, mayonnaise, sour cream, and Worcestershire sauce in a blender until creamy and pale green. Taste the sauce and season with salt and pepper, if needed.