Heat 2 tablespoons of the oil in a large skillet over high heat until hot. Reduce the heat to medium; add the onion and cook, stirring occasionally, until softened, about 5 minutes. Increase the heat to medium high; add the mushrooms and cook, stirring, until the mushrooms have browned and the liquid they release has evaporated, 4 to 5 minutes. Season the mushroom mixture with salt and pepper to taste and transfer it to a bowl; add the chiles and olives and set aside. Wipe out the skillet.
Gently shape the beef into four 4-inch burgers; season with salt and pepper. Heat the remaining tablespoon of oil in the skillet over high heat until hot. Reduce the heat to medium-high, add the burgers, and cook 3 minutes. Turn the burgers and cook 2 minutes. Top each with a quarter of the chile mixture and a slice of cheese. Add the beer to the skillet; cover and steam until the cheese has melted, about 3 minutes.
Transfer the burgers to the toasted buns and serve.