Place the arugula, scallions, basil, and salt and pepper in a food processor or blender and pulse to chop, then run the machine to chop finely.
In a medium bowl combine the mayonnaise, sour cream, lemon juice and Worcestershire sauce, if using. Turn the arugula mixture into the bowl and stir well to combine. Taste and adjust the seasonings.
Place the meat in a bowl and add salt and pepper. Use your hands to lightly but thoroughly mix the seasonings into the meat. Divide the beef into 4 even portions, and use your hands to form even, round patties, about 3/4-inch thick. Use your fingers to press a small indentation into the middle of each patty.
Make the burgers: Place the patties on the grill and cook for 3 to 4 minutes on each side, until they are cooked to your liking. During the past minute of cooking sprinkle the cheese over the burgers, distributing them evenly. Also place the buns on the grill and let them toast lightly, if desired.
Place a cheese covered patty on the bottom of each bun, and slather the of top of each bun with the sauce. Sprinkle over the onions, if desired. Place the bun top over the burger and serve.