Heat the olive oil in a large heavy-bottomed pot over medium heat. Add diced onions and minced garlic, sautéing until fragrant and translucent, about 5 minutes. The onions should be soft and slightly golden.
Add diced carrots and celery to the pot, stirring well. Cook for about 4 minutes until they start to soften and release their aroma, and the vegetables are slightly tender but still vibrant in color.
Pour in the chicken broth and bring it to a gentle simmer. Let it bubble softly for about 10 minutes, allowing the flavors to meld and the vegetables to cook through.
Stir in the shredded cooked chicken and rice. Reduce the heat to low and let the soup simmer gently for about 15 minutes, stirring occasionally, until the rice is tender and the flavors combine beautifully.
Taste the broth and add a squeeze of fresh lemon juice to brighten the flavors. Adjust with a pinch of salt and pepper if needed.
Once the soup is flavorful and the rice is cooked through, remove from heat. Drizzle the toasted sesame oil over the top for a nutty aroma and gentle sheen.
Ladle the hot soup into bowls, garnishing each with chopped scallions for a fresh, crisp contrast. Serve immediately and enjoy the bright, comforting flavors.