PREHEAT THE GRILL: Heat the grill 450 to 500 degrees F.
BURGERS: In a large mixing bowl, add the beef straight from the fridge, the minced garlic, green onions, and Worcestershire sauce. Fold the meat just a few times (no longer than for 30-40 seconds) until ingredients are incorporated. Be careful to not overwork or overhandle the meat.
SHAPE: Line a large sheet pan with two layers of parchment paper. Shape the burger patties on the tray instead of with your hands (this helps keep them from being over handled while also providing leverage to get the patties firm). Divide all the meat into six equal parts and then separate each part into two parts (with 1 part larger than the other). Gently form each large portion into a patty that is about 4 to 4 and 1/2 inches across and with 1-inch sides. Make a shallow depression in the middle. Add 2 tablespoons grated cheese into each depression. Cover the cheese with the smaller patty, connecting the beef together and forming into a patty with 1 inch thick sides. Again, depress the center of the patty (see pictures in the post). The depression keeps the patty from shrinking too much and puffing up too much. Cover with plastic wrap and place in the fridge until the grill is completely heated.
BUNS: Melt the butter and then brush (with a pastry brush) over both sides of all the buns. Set aside.
GRILL BURGERS: Prepare the fully preheated grill by cleaning with a grill brush. Generously oil the grill (I drench a rolled-up paper towel in vegetable oil and, holding it with tongs, rubbing it over the grill grates). Remove burgers from the fridge and generously salt and pepper both sides of the patties. Hold your hand 6-7 inches above the meat and sprinkle the seasoning liberally or to taste. In total, I use about 1 and 1/2 teaspoons each of salt and pepper to all the patties.Place patties INDENT SIDE DOWN on the grill. Cook to desired doneness:Rare burgers: 125 degrees F (about 4 minutes total) Medium-rare burgers: 135 degrees F (about 5 minutes total) Medium burgers: 145 degrees F (about 6-7 minutes total) Well-done burgers: 160 degrees F (about 8 minutes total) (We like medium rare best) Flip the burgers once during cooking, at the halfway mark. Never press down on the patties (this will make fat and flavor escape). Use a wide metal spatula for best results in flipping and removing patties.
GRILL BUNS: When you have about 1 minute left of the cooking time for the patties, add the buttered buns to the grill (over indirect heat, alongside the edges of the grill). Watch carefully; they are toasted in about 30 seconds to a 1 minute.
SERVE: Remove burgers and buns from grill and transfer to a clean platter. Let the burgers rest for 1-2 minute before serving. Assemble by adding burger sauce (See Note 1) and your favorite veggies such as lettuce, tomatoes, pickles, red onion, etc.