In a large, heavy-bottomed skillet or Dutch oven, heat 1/4 cup extra-virgin olive oil over medium heat and add the onion. Cook for 5 minutes, until it is softened and golden.
Season the lamb chunks with salt and pepper and add to the skillet. Increase the heat to high and sear the lamb on all sides, working in batches if necessary to avoid overcrowding the pan. Stir in the wine, scraping the bottom of the pan with a wooden spoon to incorporate the browned bits of meat and onion, and season with salt and pepper. Cook over medium-low heat for about 90 minutes, until the meat is very tender. Add more wine or water, if necessary, to keep the pan from drying out.
When the meat is cooked, combine the cheese, eggs, and bread crumbs in a large bowl and whisk together. Season with salt and pepper and add the roasted pepper strips. Increase the heat under the lamb to high and add the egg mixture to the pan. Stir as for scrambled eggs and cook until the eggs harden. Serve immediately.