Step 1: Place the ground beef, cheese, and scallions in a mixing bowl and mix well with a wooden spoon.
Step 2: Wet your hands with cold water and divide the meat into 4 equal portions. Working quickly and with a light touch, pat each portion into a ¾-inch thick patty. Make a slight depression in the center. Arrange the burgers on a plate lined with plastic wrap and refrigerate until you’re ready to grill them.
Step 3: While the burgers are chilling, make the Made-From-Scratch Ketchup, if desired.
Step 4: Set up the grill for direct grilling and preheat to high. Brush and oil the grill grate. Generously season the burgers with salt and pepper on both sides, then place on the hot grate. Grill the burgers until cooked through, 4 to 6 minutes per side for medium. To test for doneness, insert an instant-read meat thermometer though the side of a burger into the center: The internal temperature should be about 160°F for medium.
Step 5: Meanwhile, brush the buns with the melted butter and toast them on the grill, 1 to 2 minutes per side.
Step 6: Place a lettuce leaf on each bun. (This keeps the bottom of the bun from getting soggy.) Add the burgers, tomato, pickle, and onion slices (if using). Slather on the ketchup and dig in. Yes, there should also be plenty of cold beer.
