Start by heating a large heavy-bottomed pot over medium heat and add a tablespoon of oil. Once shimmering, add diced onion and minced garlic, cooking until fragrant and translucent, about 5 minutes.
Add chopped carrots and celery to the pot, stirring and cooking until they begin to soften and release their aroma, about 4 minutes.
Pour in the chicken broth and bring the mixture to a gentle simmer, letting it cook for about 10 minutes so the flavors meld and the vegetables soften further.
Stir in the shredded cooked chicken and rice, then reduce the heat to low and let everything simmer gently until the rice is tender and the soup thickens slightly, about 15 minutes. Stir occasionally to prevent sticking.
Taste the soup and squeeze in fresh lemon juice, adjusting the amount to brighten and balance the flavors without overpowering.
Finish by drizzling a teaspoon of toasted sesame oil over the soup for a nutty aroma, then sprinkle chopped scallions on top for a fresh burst of flavor.
Serve the soup hot, with a spoonful of scallions in each bowl, and enjoy the comforting, bright flavors in every spoonful.