In a large bowl, combine the chicken, chorizo, garlic, oregano and salt. Form the mixture into 4 equal sized patties, about the same width as the bread that you will be using. (My patties were more oval shaped than circle because I used a circle flatbread that gets folded in half for the sandwich. The patties will be thin.) The mixture is a bit sticky, so if needed, wet your hands so that the mixture doesn’t stick as badly. Place the patties on a plate, then cover and refrigerate for 30 minutes.

In a large sauté pan over medium heat, melt the butter. Add the oil and the onions, then season with salt and pepper. Sauté until the onions are caramelized, about 35 minutes.

Preheat a griddle over medium-high heat. Add 2 tablespoons of the oil, then cook the patties until cooked through and crispy, about 3 minutes on each side. Remove the patties to a plate, then wipe down the griddle.

Assemble the sandwiches by placing one slice of cheese on the bottom half of each flatbread, followed by a patty, another slice of cheese, and then 1/4 of the onions. Fold the flatbread over, forming a sandwich. Brush the remaining 2 tablespoons of oil on the griddle and grill each sandwich until the cheese is melted and the bread is toasted, about 2 minutes on each side.

Serve immediately.