Heat a tablespoon of oil in your 4-quart pot over medium heat until shimmering.
Add diced onion and sauté until translucent and fragrant, about 5 minutes, listening for a gentle sizzle and watching for the onions to turn clear.
Stir in minced garlic, sliced ginger, and bruised lemongrass stalk, cooking until aromatic, about 2 minutes, filling the kitchen with a bright, herbal scent.
Pour in the chicken broth and bring the mixture to a gentle simmer; let it cook for about 20 minutes, allowing the flavors to meld and the broth to develop depth.
Remove the lemongrass stalk, then add the shredded cooked chicken to the broth, stirring to combine, and cook for another 10 minutes to warm through.
Finish by squeezing fresh lime juice into the soup and stirring in chopped cilantro, adjusting salt to taste for a balanced, bright flavor.
Ladle the hot soup into bowls, garnish with extra cilantro or lime if desired, and serve immediately for a fragrant, comforting meal.