Cut off and discard the root end and the dark green leaves of the leek. Then rinse the remaining leek well, and cut in half lengthwise. Slice into thin half-moons. (If there was a lot of dirt between the layers of the leek, put the sliced leeks in a bowl of cold water, swirl, then let sit for a few minutes to allow the dirt to settle. Then remove leeks with a slotted spoon and pat dry.)
Peel carrots, then chop into small ¼-inch pieces.
Peel ginger and chop in half.
Heat butter and oil over medium heat in a large saucepan or dutch oven until butter melts.
Add sliced leeks, chopped carrots, and ginger. Sauté until softened, about 8-10 minutes over medium heat, stirring frequently.
Add chicken broth, 2 tablespoons of lemon juice, chicken, rice, and salt.
Bring soup to a low simmer, then cook for 15 minutes, or until rice is cooked through and carrots are just tender.
Whisk egg in a small bowl until yolk and white are well-combined, then pour into soup while stirring.
Season soup with additional salt and pepper, and add remaining tablespoon of lemon juice, if desired.
Remove ginger chunks before serving. Garnish with chopped parsley, if desired.
Notes
Make-Ahead Instructions: You can make this soup in advance, but keep in mind that rice can plump up on you the longer it sits in the liquid. Some types of rice plump more than others, such as jasmine rice. If you don’t want to run the risk of the rice swelling too much, you can also make the soup and the rice separately, then add the cooked rice just before you are ready to serve. Soup will last in the refrigerator for 3-5 days in an airtight container. You can also freeze this soup, though because of the rice plumping, it's recommended to freeze the soup without the rice and add later when you defrost and reheat it.Poaching Chicken Instructions: If you don’t have cooked chicken, here are some easy instructions for poached chicken. Put two boneless, skinless chicken breasts in a large saucepan, and cover with water. Add any aromatics you have around, such as bay leaves, garlic cloves, fresh herbs, peppercorns, celery, or carrots. You could throw in some of those (washed) dark green leek leaves too. You might also add a little white wine. Bring liquid just to a boil, then turn down to a simmer. Cover and cook for 8-12 minutes, until they are fully cooked to 165 degrees Fahrenheit. Let cool, then chop or shred with a fork or your fingers.