Heat cast iron skillet over medium heat. Brush onion slices with a bit of olive oil and cook in skillet until slightly charred on both sides, about 3-4 minutes a side. Lightly toast buns in skillet or in toaster oven.
While onions are cooking, combine mushrooms, garlic and black beans in the food processor and pulse until mixture is coarsely chopped. Do not pulverize! You want a little texture, not a paste.
Add Panko, oregano, cumin, salt, chile powder and egg and pulse just until combined. Repeat: Do not pulverize! You want a little texture, not a paste. Shape into four 1/2 inch thick patties.
Lightly oil the skillet and cook the patties approximately 12 minutes, turning once. Add cheese to the top of the burgers the last two minutes, cover the pan to assist melting.
Place patty on toasted bun, top with onion slice and a half chile slice (or more if desired).