Process Hellmann's® or Best Foods® Real Mayonnaise, green chilies and garlic in food processor until smooth. Remove to medium bowl, then stir in jalapeno peppers and cilantro. Season, if desired, with salt and black pepper. Cover and refrigerate at least 30 minutes.
Meanwhile, shape ground beef into 4 burgers. Brush each burger on both sides with oil, then season, if desired, with salt and black pepper. Grill, turning once, 8 minutes or until desired doneness. Top burgers with cheese and cook until cheese is melted. Arrange burgers on buns, then top with a dollop of green chili mayonnaise. Serve remainder of mayonnaise in small bowl for dipping.