Combine mushrooms and onion in a food processor and pulse to finely chop.
Heat 2 tbsp of the oil in a large cast-iron skillet over medium-high heat.
When oil is shimmering hot, add the mushroom mixture plus ½ tsp of the salt.
Cook, stirring occasionally, until the mushrooms have lost most of their moisture and are starting to brown, about 10 minutes, then scrape into a heat-safe bowl to cool slightly.
Stir ground beef into the mushroom mixture along with ½ tsp more salt, Worcestershire sauce, and pepper.
Use your hands to form four patties about ¾” thick.
Wipe out the skillet and return it to medium-high heat with the remaining 1 tbsp of oil.
When shimmering hot, add the burgers and cook undisturbed until they flip easily with a spatula, about 4 minutes.
Flip and continue cooking on the other side until browned, about 4 minutes more.
Top burgers with cheese and cover.
Cook to an internal temperature of 165ºF.
To assemble burgers, divide mayonnaise among the buns, add the burgers, and top with greens.