Heat olive oil in a large heavy-bottomed pot over medium heat until shimmering and fragrant.
Add chopped onions, carrots, and celery to the pot. Cook, stirring occasionally, until the vegetables soften and release a sweet aroma, about 8 minutes, and the vegetables are slightly translucent.
Place the chicken thighs into the pot, skin-side down if skin-on, and brown each side for about 5 minutes until golden brown and fragrant.
Pour in the chicken broth, add the sprig of rosemary, and bring the mixture to a gentle simmer. Cover the pot and cook for 20 minutes, allowing the flavors to meld and the chicken to become tender.
Remove the chicken thighs from the pot, shred the meat with two forks, discarding the bones if bone-in, and return the shredded chicken to the soup.
Season the soup generously with salt and pepper, then add the lemon juice to brighten the flavors. Stir well and let it simmer uncovered for an additional 10 minutes.
Taste the soup and adjust seasoning as needed, adding more salt, pepper, or lemon juice for balance.
Serve the soup hot, garnished with chopped parsley and a drizzle of good olive oil for extra richness and freshness.
