In a food processor, pulse the ingredients for the chimichurri until a chunky sauce forms (10 to 12 pulses); season with salt.
In a small bowl, toss the onion and vinegar to make quick pickles; season with salt and pepper.
In a medium bowl, mix the beef and Worcestershire; season. Form 4 patties (thinner in the centers for even cooking).
Heat a large griddle or cast iron skillet over medium-high to high. Drizzle the patties with oil. Add to the griddle. Cook, turning occasionally and pressing down with a spatula to help create a crispy crust, 7 to 8 minutes. Transfer the burgers to a plate and cover with foil to keep warm. If necessary, scrape the griddle to remove any browned bits of beef. Add the 4 slices of cheese to the griddle, spacing them an inch or two apart. Cook until melted and lightly crispy on the bottom, 1 to 2 minutes.
Uncover the burgers and top with some chimichurri sauce. Using a thin metal spatula or two, transfer the fried cheese to the burgers.
Top the roll bottoms with some pickled onions, the chimichurri cheeseburgers, lettuce, tomato slices and roll tops.