Heat a 12-inch cast-iron skillet over medium-high and heat broiler, with rack in top position. Cook bacon in skillet until fat renders and bacon is browned, about 2 minutes on each side. Transfer bacon to paper towels to drain and break in half; reserve fat in skillet and return to heat.
Form beef into 4 patties, roughly 5 inches in diameter. Season with salt and pepper and place in skillet. With a metal spatula, press down gently to flatten each against skillet. Reduce heat to medium. Cook 3 minutes, then flip, scraping against skillet with spatula so burgers release cleanly. Cook 3 minutes for medium-rare, then top each with 1 slice cheese. Broil until cheese begins to melt, 1 minute. Transfer burgers to a plate and keep warm.
Pour off fat from skillet and reserve. Add onion and cook over medium-high, 30 seconds on each side; transfer to plate. Add 2 tablespoons reserved fat to skillet and toast half the buns, cut side down, until golden, 2 minutes. Repeat with remaining buns.
On bottom halves of buns, layer onion, burgers, bacon, and escarole. Top with remaining halves of buns and serve immediately.
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