Warm the pot. Set a large soup pot or Dutch oven over medium heat.
Add olive oil or butter and let it melt until shimmering.
Sauté the aromatics. Add onion, carrots, and celery with a pinch of salt. Cook 6–8 minutes, stirring occasionally, until softened and lightly golden at the edges. Add the garlic and cook 30 seconds, just until fragrant.
Build the base. Stir in the potatoes, bay leaf, and thyme.
Pour in the chicken broth and bring to a gentle boil.
If using raw chicken, add it now. Nestle bone-in chicken pieces into the pot. Reduce heat to a simmer, cover partially, and cook 25–30 minutes until the chicken is cooked through and the potatoes are tender. Remove the chicken to a plate, cool slightly, then shred, discarding skin and bones.
If using cooked chicken, wait. Simmer the vegetables 15–20 minutes until the potatoes are tender before adding the shredded chicken.
Add the greens and finish. Stir in shredded chicken and peas (or green beans).
Simmer 3–5 minutes to warm through. Remove bay leaf and thyme stems.
Brighten and season. Squeeze in the lemon and add parsley. Taste and season with salt and plenty of black pepper.
The lemon should lift the broth without making it sour.
Optional starch. If adding pasta or rice, stir it in now. For pasta, cook it separately to al dente, then add to prevent sogginess later.
Serve. Ladle into warm bowls and finish with a drizzle of olive oil, extra parsley, or a crack of pepper. Crusty bread on the side is never a bad idea.