Soften the vegetables: Melt the butter in a large Dutch oven or soup pot over medium heat. Add the onion, carrots, and celery. Cook, stirring occasionally, until the vegetables begin to soften, about 7 to 8 minutes. Add the garlic and cook for 30 seconds.
Make the roux: Sprinkle the flour over the vegetables. Cook, stirring constantly, for 1 to 2 minutes. The flour should coat the vegetables but should not brown.
Add the broth and potatoes: Gradually stir in the chicken broth, scraping up any browned bits from the bottom of the pot. Add the potatoes, poultry seasoning, thyme, salt, and pepper.
Simmer until tender: Bring the soup to a gentle boil. Reduce the heat to low and simmer, uncovered, for about 12 to 15 minutes, stirring occasionally, until the potatoes and carrots are tender.
Stir in the chicken, peas, and cream: Add the cooked chicken, frozen peas, half-and-half, and parsley. Cook over low heat for about 5 minutes, just until the chicken and peas are warmed through. Do not let the soup boil after adding the cream.
Season and serve: Taste and add more salt and pepper, if needed. Ladle into bowls and serve with warm biscuits.