Heat a grill to medium-high direct heat and lightly oil the grates using a paper towel soaked in canola oil or nonstick grilling cooking spray.
Spread a 16-inch-long piece of nonstick aluminum foil on a cutting board (or mist a piece of regular aluminum foil with nonstick cooking spray). Top with the onions, 1/4 teaspoon each salt and pepper and 1 tablespoon water. Bring two sides of the foil together, fold over twice to create a seal and then seal the remaining two ends. Put on the grill and cook for 12 minutes, flipping once. The onions should be tender with only a bit of charring. Carefully open the top of the foil to let all the steam out and continue to cook the onions, tossing, until browned and lightly charred, 3 minutes longer. Remove from the heat.
While the onions cook, mix the ketchup and mustard together in a small bowl. Season with pepper and set aside.
Lightly toast the English muffins on the grill and place one top and bottom on each of 4 plates. Spread some of the ketchup sauce on the bottom half of each muffin and top with a slice of tomato.
Divide the beef into 4 patties about 4 inches in diameter and just under 1/2-inch thick. Sprinkle with a total of 1/2 teaspoon salt and 1/4 teaspoon pepper. Place the burgers on the grill. Cook until the edges of the burgers are browning, 3 to 5 minutes, and then flip. Brush the burgers with the remaining ketchup sauce and evenly top with the shredded cheese. Cover the burgers with the grill lid or a large heatproof bowl and continue to cook until the cheese has melted and the burgers are still a bit pink on the inside, about 3 minutes more.
Put one burger on top of the assembled muffin bottom and top with some of the grilled onions and the top of the muffin. Serve with pickles if desired.