How to Make It
Shape ground beef into 4 thin patties to match the size and shape of bread slices. Set aside.
Toss together scallions, oil, a pinch of salt, and a pinch of black pepper in a bowl. Heat a large skillet over medium-high. Add scallions; cook, turning occasionally, until charred in spots and tender, about 4 minutes. Transfer scallions to a cutting board, and coarsely chop. Stir together chopped scallions, 1/4 cup mayonnaise, chipotle chiles, and adobo sauce in a small bowl; set aside. Wipe skillet clean.
Add butter to skillet; melt over medium-high. Add beef patties, and sprinkle evenly with 1/2 teaspoon salt and 1/2 teaspoon black pepper. Cook patties, pressing tops occasionally using a spatula to keep patties thin and flat, until bottoms are well browned, about 2 minutes. Flip patties; sprinkle evenly with remaining 1/2 teaspoon salt and remaining 1/2 teaspoon black pepper. Cook patties, pressing tops occasionally, until a crust forms on bottoms and beef is cooked to desired degree of doneness, 2 to 3 minutes for medium-rare (about 145°F). Transfer patties to a plate lined with paper towels. Wipe skillet clean.
Spread about 1 tablespoon scallion-chipotle mayonnaise on 4 bread slices. Top each with 1 cheese slice, 1 patty, and 1 additional cheese slice. Cover with remaining bread slices.
Return skillet to heat over medium. Spread 1 outer side of each sandwich with 1/2 tablespoon mayonnaise; place sandwiches, mayonnaise side down, in skillet. Cook sandwiches, pressing tops with a spatula, until bottoms are golden, 1 minute and 30 seconds to 3 minutes. Spread top of each sandwich with 1/2 tablespoon mayonnaise; flip sandwiches. Cook sandwiches, pressing tops occasionally with spatula, until bottoms of sandwiches are toasty and cheese is melted, 2 to 3 minutes. Cut sandwiches in half, and serve.