Zest the 2 lemons into a small bowl, stopping before the white pith, then juice them into a measuring cup and set aside about ¼ cup (60 ml). Cut the additional lemon into wedges for serving.
Pat the chicken dry, trim any large fat pieces, and cut into ¾-inch (2 cm) pieces. Toss with ¾ teaspoon salt and ¼ teaspoon pepper. Dice the onion, carrots, and celery into ¼-inch (6 mm) pieces, mince the garlic, and chop the parsley.
Heat 1 tablespoon olive oil in a Dutch oven over medium-high heat. Add the seasoned chicken in a single layer (brown in two batches if crowded) and cook 4 to 5 minutes, stirring occasionally, until golden outside. Transfer the chicken and juices to a plate, leaving the browned bits in the pot.
Reduce the heat to medium and add the remaining 1 tablespoon olive oil, the onion, carrots, celery, and remaining ¼ teaspoon salt. Cook about 5 minutes, stirring often and scraping up the browned bits, until the onion is translucent and the carrots begin to soften without taking on color.
Add the minced garlic, dried oregano, and dried thyme and stir constantly for about 30 seconds, until fragrant. Do not let the garlic brown, because it turns bitter quickly.
Pour in the chicken broth and add the bay leaves and remaining ¼ teaspoon black pepper. Return the browned chicken and any juices to the pot, bring to a boil (about 4 minutes), then reduce to a gentle simmer, partially cover, and cook 10 minutes until the carrots are nearly tender.
Stir in the egg noodles, pushing them under the broth, and simmer 6 to 8 minutes, stirring occasionally so they do not clump, until tender. Confirm the chicken has reached 165°F (74°C), then take the pot off the heat as soon as the noodles are done.
Off the heat, discard the bay leaves. Stir in the lemon zest, 3 tablespoons of the lemon juice, and the parsley. If using spinach, add it now and stir about 1 minute until wilted. Taste and add the remaining lemon juice 1 teaspoon at a time, plus a pinch of salt if needed, until bright but balanced.
Ladle the soup into bowls and serve right away with lemon wedges on the side.