Preheat you barbecue grill to 400-450°F over direct heat.
Mix the melted butter and garlic in a small bowl. Use a brush to apply the garlic butter to both sides of each piece of bread. Arrange the pieces of bread, two at at time, at an angle (for nice grill marks) on the grill grate directly over hot coals. Cook for 20-30 seconds then flip and cook the other side for another 20-30 seconds per side. Remove the bread and set aside.
Mix the burger sauce ingredients in a small bowl.
Divide the ground beef into 4 patties, each weighing ½ pound. Press the patties flat to about ½" thick and 6" wide using a burger press. Grill the burgers over direct heat for 3 minutes. Flip them over, cover each patty with 2 pieces of cheese, and turn off the grill. Cook for another 2 minutes.
Spread the burger sauce on one side of each piece of toast. Assemble the burgers and enjoy.