Dice the onion, slice the carrots and celery, mince the garlic, and shred the cooked chicken. Measure the remaining ingredients before heating the pot. Keep the biscuit dough refrigerated until the soup is ready.
Melt the butter in a large Dutch oven over medium heat. Add the onion, carrots, and celery. Cook for 6 to 8 minutes, stirring often, until the onion becomes translucent and the vegetables begin to soften.
Stir in the garlic, poultry seasoning, dried thyme, salt, and black pepper. Cook for 30 seconds, stirring constantly, until the garlic smells fragrant.
Sprinkle the flour evenly over the vegetables. Cook for 1 minute while stirring and scraping the bottom of the pot. The flour should coat the vegetables and lose its raw appearance.
Slowly pour in 1 cup of chicken broth while whisking. Once the mixture becomes smooth, gradually whisk in the remaining broth. Add the milk and whisk until no dry flour remains.
Bring the mixture to a gentle simmer over medium heat. Cook for 5 to 7 minutes, stirring often, until the broth lightly coats the back of a spoon. Do not let it reach a hard boil.
Stir in the shredded cooked chicken. Reduce the heat to medium-low and cook for 2 minutes, allowing the chicken to warm through.
Remove the biscuit dough from the refrigerator and separate the biscuits. Cut each biscuit into 4 equal wedges using a knife or kitchen scissors. Do not flatten or stretch the dough.
Place the biscuit pieces in a single layer over the gently simmering soup. Leave a small amount of space between the pieces and avoid stacking them. Sprinkle the frozen peas into the open spaces.
Cover the pot with a tight-fitting lid and reduce the heat to low. Cook for 15 minutes without lifting the lid. The trapped steam is needed to cook the tops and centers of the dumplings.
Cut through the largest dumpling. Its center should look fluffy and bread-like, with no wet or shiny dough. If it is still raw, cover the pot and cook for another 3 to 5 minutes.
Remove the pot from the heat and let the chicken and dumplings rest uncovered for 5 minutes. Taste the broth and add more salt or black pepper if needed. Spoon into bowls and garnish with chopped parsley.
