Place the ground beef in a shallow bowl. Add the salt, a few grinds of black pepper, mayonnaise, and panko (see notes).
Use a wooden spatula to combine the meat ingredients as best you can before doing a final, gentle mix with your hands.
Gently form the patties into quarter-pound portions and make a slight depression in the center of each with your thumb (see notes). Transfer to a plate and chill for at least 15 minutes before grilling.
Preheat a gas or charcoal grill to a medium-high temperature (about 400°F) and oil the grates.
Add the onion slices and grill until lightly browned, 1-1/2 minutes per side. Transfer to a plate and set aside.
Add the burgers and grill for approximately 4 minutes on the first side. You'll know when it's time to turn them because juices will begin to bead on the top of each burger indicating that they are cooked through the center.
Use a spatula to turn them and grill for an additional 2 to 3 minutes on the second side, adding the pimento cheese after about 1 minute. Transfer the burgers to a plate and allow them to rest for 5 minutes before serving.
To serve, place a fried green tomato on the bottom of each bun. Top with an onion slice and a burger.
Tips for Making This Recipe
Adding mayonnaise and panko helps to hold the juices in the burger during cooking without changing the flavor or texture of the beef. The coarse texture of the panko is important because, unlike dry breadcrumbs, they don't add density to ground beef.
Making a depression in the center of each burger helps the burgers maintain an even thickness rather than developing a thicker "dome" in the center.