Cut skirt steaks into about 3 or 4 smaller pieces so it's easier to fit them into marinade dish.

Whisk orange juice, lime juice, olive oil, garlic, salt, cumin, pepper, oregano, and cayenne pepper together in a large bowl. Place skirt steak pieces, 1 at a time, into the marinade to thoroughly coat them. Add sliced onions and toss with the meat.

Transfer mixture and marinade to a resealable plastic bag. Squeeze out air, seal bag, and place on a dish. Refrigerate 2 to 3 hours.

Transfer pieces of meat onto paper towel-lined rimmed sheet pan to drain for a couple of minutes.

Cook over hot coals. Grill first side 3 to 4 minutes. Turn and grill second side until internal temperature is about 125 degrees F (53 degrees C). Look for a shiny glossy surface indicating the meat juice is coming to the surface; this indicates the meat is just about done. Transfer to a plate and allow meat to rest a few minutes.

Slice into 1/2-inch slices and arrange on a serving plate. Spoon accumulated juices over the meat. Drizzle with olive oil and sprinkle with coarse salt and chopped cilantro. Serve with lime wedges.