Heat the olive oil (or butter) in a large stock pot or dutch oven over medium-high heat. Season the chicken lightly with salt. Add the chicken to the pot and sear for about 2 minutes per side. Remove and set aside. Add the onion and carrots to the pot and sauté for about 3 minutes, scraping up any browned bits. (Optional: deglaze with a splash of dry white wine or broth to help loosen those brown bits) Add the celery and cook for 2 more minutes, then stir in the grated garlic and cook for 1 minute until fragrant.
Pour in the chicken broth, add the bouillon cube, Italian seasoning, and salt. Return the chicken to the pot. Cover and bring to a gentle simmer. Cook for about 10 minutes, or until the chicken reaches an internal temperature of 165°F.
Remove the chicken, shred it with two forks, and return it to the soup.
Add the noodles, cover, and continue simmering until the noodles are al dente, about 8 minutes.
Finish with lemon juice, taste, and adjust seasoning as needed. Serve hot.