Melt 2 Tbs. butter in a dutch oven over medium heat and add the chicken. Season with the 2 tsp. salt and 1/2 tsp. pepper. Cook the chicken until well browned and cooked through; set aside to cool before shredding.
Add the remaining stick of butter to the dutch oven and allow it to melt before adding in the carrots, celery, and onion. Stir occasionally with a wooden spoon until tender, about 12-15 minutes (I find carrots take the longest, just keep testing them until tender). Add in the garlic and poultry seasoning; stir the entire mixture together for about 1 minute.
Stir in the 1/2 cup + 2 Tbs. flour, stirring constantly for 1 minute; stir in the wine (or substituted stock), scraping the bottom of the dutch oven as you do.
Add the 2 quarts of chicken stock along with the bay leaves. Turn up the heat and bring to a boil; reduce heat and simmer for about 15 minutes. Remove the bay leaves.
Add the shredded cooked chicken and peas along with the heavy cream. Simmer until thickened, 8-10 minutes.
Finish off the soup with plenty of salt and pepper to taste.
Enjoy with fresh bread, rolls, or biscuits.