Heat olive oil in a large pot over medium heat.
Add diced onion, carrots, and celery to the pot and sauté for 5-7 minutes until vegetables are softened.
Add minced garlic and cook for an additional 1 minute until fragrant.
Pour in chicken broth and bring to a boil.
Reduce heat to low and add shredded chicken, dried thyme, dried parsley, and dried rosemary.
Simmer the soup for 15 minutes, stirring occasionally.
Season with salt and black pepper to taste.
Remove the pot from heat and allow the soup to cool slightly for 5 minutes.
Divide the spiralized zucchini noodles evenly among the 4 glass jars.
Ladle the hot soup with chicken and vegetables over the zucchini noodles in each jar, filling to about 1 inch below the rim.
Allow the jars to cool to room temperature before sealing with lids.
Store the jars in the refrigerator for up to 4 days.
To serve, heat the soup jar contents in a microwave-safe bowl or pour into a pot and warm on the stove until the zucchini noodles are tender, about 3-5 minutes.