To make the sauce: Place the mayonnaise, sour cream, basil, scallion, garlic, Worcestershire sauce, salt, and pepper in the bowl of a food processor and puree until creamy and pale green. Transfer to a small bowl, cover with plastic wrap, and refrigerate until you are ready to use it.
To make the burgers: Heat a charcoal or gas grill to medium-high. Divide the ground beef into 4 equal pieces and gently form into patties. Use your thumb to make a small indentation in the middle of each burger (this is so the burger grills flat and doesn't contract and puff up on the grill). Brush both sides of the burgers with the oil and then season with salt and pepper.
Place the burgers on the grill and cook until they have grill marks, about 4 minutes. Flip the burgers, cook 2 minutes longer, and then top each with about 1 tablespoon of the mozzarella. Cook the burgers 2 more minutes for medium-rare doneness and remove from the grill. Place the buns on the grill, cut side down, and lightly toast, 30 seconds to 1 minute.
Set a burger on each bottom bun half. Top with a salted tomato slice. Spread 1 tablespoon of the garlic-herb sauce on the top bun half, sprinkle with Parmesan, cover the burger, and serve with more sauce on the side for dipping.
Reprinted from the book Ten Dollar Dinners. Copyright (c) 2012 by Melissa d'Arabian. Published by Clarkson Potter, a division of Random House, Inc.
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