Heat 1 tablespoon of the oil in a large nonstick skillet over medium-high heat. Add the eggs and a generous pinch of salt and cook, swirling the pan, until set on the bottom, about 2 minutes. Fold the omelet over itself and cook until just golden and set, about 2 minutes more. Slide the omelet onto a cutting board, let cool slightly and then dice.
Meanwhile, whisk the soy sauce and lime juice together in a small bowl.
Heat the remaining 3 tablespoons oil in the skillet over medium-high heat. Add the pepperoni and stir-fry until rendered and just starting to crisp, about 2 minutes. Add the ginger and garlic and stir-fry until very fragrant and lightly browned, about 30 seconds. Add the rice and stir-fry until coated in the fat and hot, 2 to 3 minutes. Stir in the soy mixture, then fold in the eggs, cilantro and scallions. Season with salt and pepper. Serve hot with sambal oelek if using.