Use a paper towel to pat the chicken dry. Season with salt and pepper.
In a 5-quart dutch oven or soup pot, heat the oil over medium-high heat. Add the chicken and brown on both sides, 5 to 8 minutes per side. Reduce the heat if the pan starts to scorch. Remove the chicken from the pan and set on a plate.
Add the carrots, onion, and celery and 1/2 tsp of sea salt to the pan. Cook over medium-low heat until they begin to soften, about 5 minutes. Add the garlic and cook for a minute more.
Pour in the chicken stock and stir, scraping up any brown bits on the bottom of the pan. Add the chicken back to the pot and bring to a simmer. Stir in the Italian seasoning and thyme. Reduce heat to low, cover and let cook until a meat thermometer placed into the thickest part of the breast reads 160 to 165 degrees, 10-15 minutes.
Take the chicken breasts out of the pot and place onto a clean plate or cutting board. Let cool enough to where you can handle it, then remove the skin, and shred the chicken into bite-size pieces.
Bring a separate pot of water up to a boil and cook the pasta according to package directions until al dente, drain and set aside. While the chicken is cooling, bring the pot of soup back up to a simmer, add the chopped kale and let cook until kale is tender, 10-15 minutes.
Add the shredded chicken back to the pot and cook until heated through, about 2 minutes.
Add the noodles to soup and stir together. (If you know there will be leftovers, do not add the noodles to the pot. Place some pasta into your bowl, then ladle the soup over it.) Stir in the parsley, check seasoning and add salt and pepper if necessary, then serve.