In a medium saucepan, combine the chicken breasts, unsalted chicken broth, fresh thyme, fresh oregano, rosemary stalk, garlic cloves, kosher salt, and black pepper. Bring to a boil over medium-high heat, then reduce the heat to low and let it simmer gently for about 1 hour.

While the chicken is cooking, heat olive oil in a large pot over medium heat. Add chopped yellow onion and sauté for 10-15 minutes until caramelized and golden brown. Stir occasionally to prevent sticking, then add chopped carrots and celery, cooking until they begin to soften.

Once the chicken is fully cooked, remove it from the pot and allow it to cool slightly before shredding into bite-sized pieces using two forks. Strain the simmering broth to remove herbs and garlic.

Add the shredded chicken back into the sautéed vegetables and pour in the reserved chicken broth along with the strained cooking liquid. Stir well to combine and adjust seasoning with salt and black pepper if necessary.

Bring the mixture to a gentle simmer over medium heat until the vegetables are tender, about 5-10 minutes. Stir in the fresh parsley, dill, and basil until evenly distributed.

Serve the soup hot with your choice of cooked pasta, rice, or a side of crusty bread. Optionally sprinkle with freshly ground black pepper before serving.