Marinate the Steak: Preheat the grill to 425°F, using indirect heat. In a food processor, blend ¼ cup olive oil, kosher salt, black pepper, smoked paprika, cumin, garlic, oregano, and cayenne for 30 seconds until smooth. Place the skirt steaks in a bowl or resealable storage bag, pour the marinade over the steaks, and rub it in to coat evenly. Set aside for 20 minutes at room temperature.
Grill the Onions: Toss the red onion slices in the remaining 2 tablespoons of olive oil. Grill the onions over direct heat for 3-4 minutes until tender, moving them to indirect heat if they cook too fast. Set aside.
Grill the Steak: Place the marinated steaks on the grill and cook for 2 minutes on each side over direct heat. Move the steak to indirect heat, cover the grill, and cook for another 5 minutes for medium doneness. Adjust cooking times for medium-well or well-done steaks. Remove steaks from the grill and let them rest for 5 minutes before slicing.
Make the Horseradish Mayo: While the steaks rest, mix together mayonnaise, horseradish, kosher salt, black pepper, and chopped chives in a medium bowl until smooth.
Assemble the Sandwiches: Spread a heaping tablespoon of the horseradish mayo on both sides of the toasted ciabatta buns. Layer on grilled onions, lettuce, and slices of the grilled steak. Top with the other half of the bun and serve immediately.