Preheat the oven to 375°F (190°C).
Caramelize the onions by heating olive oil in a skillet over medium heat, adding the sliced onions, and cooking until golden brown, about 15-20 minutes.
Add minced garlic and dried oregano to the skillet and sauté for an additional 2-3 minutes until fragrant.
Season the chicken breasts with salt and pepper on both sides.
Layer the mozzarella cheese and caramelized onion mixture onto each chicken breast, then roll tightly and secure with toothpicks.
Dredge each roll in beaten egg and then coat with breadcrumbs.
Place the prepared chicken rolls in a baking dish and drizzle with a little olive oil or cooking spray.
Bake in the preheated oven for 25-30 minutes, or until the chicken is cooked through and golden brown on the outside.
Remove from the oven, let rest for 5 minutes to allow juices to redistribute, and serve.
