For the caramelized onions: Heat the oil in a skillet over medium heat. Add the onions and toss to coat with the oil. Cover and slowly cook, stirring occasionally, until the onions are golden, about 10 minutes. Season with salt and pepper.
For the matzo brei: Heat the oil in a saute pan over medium heat. Add the salami and fry for 3 minutes. Soak the matzo sheets in water for 15 seconds, crumble them with your fingers and add them to the salami. Add the caramelized onions and toss with the salami and matzo. Cook for an additional 3 minutes. Add the beaten eggs and cook until the eggs are cooked through, about 5 minutes.