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Heat the oil in a separate pan over medium heat and add the onions. Cook the onions until clear, about 5 minutes. Set aside.
Remove the carrot and leek from the broth and add the rice. Bring to a boil, and then turn the heat to medium-high to simmer until the rice is cooked to al dente, about 30 minutes. Add the chicken back into the broth. Add more water if needed.
In a small bowl, beat the lemon juice and eggs together. Pour 2 cups of broth slowly into the bowl, whisking continuously. Once all the broth is incorporated, add the mixture into the pot of chicken soup and stir to blend well throughout. Add the salt and pepper. Serve hot.