In a large skillet or wok, preheat 2 tablespoons of sesame oil and add the diced onion, carrot, minced garlic and ginger. Stir fry over medium heat for about 5 minutes, or until the onion softens and the mixture becomes fragrant. Turn up the heat for the next steps:
Season the shrimp with salt and pepper. Move the onion-carrot mixture to one side of the skillet and add the shrimp to the other. Cook the shrimp over high heat for 2-3 minutes, or until fully cooked. Stir everything together and move to one side of the skillet.
Add 1 tablespoon of sesame oil to the empty side of the skillet and add the leftover rice. Stir fry for 2-3 minutes or until the rice is heated through. Add the frozen peas and stir fry for 2-3 minutes. Move the whole mixture to one side and add the remaining 1 tablespoon of sesame oil to the empty side of the skillet.
Crack the 2 eggs into the oil and scramble them with the spatula until they are mostly cooked. Stir everything together and add the soy sauce. Continue stir frying until everything is heated through and mixed well. Remove from heat and stir in the scallions and lime juice.
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