Melt 4 tablespoons of butter in a large pot over medium-high heat. Add diced onion, sliced celery, and carrots. Season with kosher salt and fresh cracked pepper. Cook for about 8 minutes until tender.
Cut refrigerated biscuit dough into 6 pieces each and toss in a thin layer of flour.
Add minced garlic, Italian seasoning, dried sage, fresh thyme (if using), and more salt and pepper to the sautéed veggies. Cook for about a minute.
Sprinkle 3 tablespoons of flour over the veggie mixture, stir, and cook for another minute. Slowly pour in chicken broth to deglaze the pan.
Add shredded rotisserie chicken, heavy cream, frozen peas, and bay leaves. Stir and bring to a simmer.
Carefully add floured biscuit dough quarters on top of the soup. Cover and reduce heat to low; simmer for about 15 minutes.
Taste the soup and season with salt and pepper as needed. Garnish with freshly chopped parsley before serving.