In a bowl, combine mayonnaise, sour cream or Greek yogurt, minced garlic, lemon juice, and dried dill. Mix well, season, and refrigerate for at least 15 minutes.

In a large mixing bowl, combine ground beef, roasted garlic, salt, black pepper, and Worcestershire sauce. Gently mix with hands, being careful not to overwork the meat.

Divide the meat mixture into four portions, shape into patties, and create slight indentations in the center.

Preheat the grill to medium-high heat. Cook the patties for 4-5 minutes on one side, then flip and cook for an additional 4 minutes or to desired doneness.

Toast the burger buns cut-side down on the grill or in a toaster for about 2-3 minutes until golden brown.

Assemble the burgers by placing a patty on each toasted bun and drizzling with the creamy garlic sauce. Serve immediately.