Prep the chicken (if using raw): Pat chicken dry and season lightly with salt and pepper. If using rotisserie chicken, shred and skip to step 5.
Sear the chicken (optional for more flavor): In a large Dutch oven or heavy pot, heat 1 tablespoon butter (or 1 tbsp oil) over medium-high heat. Add chicken and brown 1–2 minutes per side just to develop color (it does not need to be fully cooked). Remove chicken to a plate.
Sauté the aromatics: In the same pot, reduce heat to medium. Add remaining 2 tablespoons butter. Add diced onion, carrots, and celery. Sauté until onions are translucent and vegetables begin to soften, about 5–7 minutes. Add minced garlic and cook 30–60 seconds until fragrant.
Make the roux and thicken: Sprinkle 1/3 cup all-purpose flour over the vegetables and stir to coat; cook 1–2 minutes to remove raw flour taste. Slowly pour in 6 cups chicken broth while stirring or whisking to avoid lumps. Add thyme and bay leaf. Bring to a gentle simmer.
Cook the chicken through & shred: Return the seared chicken to the pot (or add shredded rotisserie chicken). Simmer until chicken is cooked through (internal temp 74°C / 165°F) — about 10–12 minutes if raw. Remove chicken, shred with two forks, then return shredded chicken to the pot.
Adjust thickness and flavor: Stir in 1/2 cup milk or cream. Taste and season with salt and pepper. If the broth is too thin, simmer uncovered a few minutes to thicken; if too thick, add a splash more broth or milk.
Add vegetables (if using frozen peas): Stir in 1 cup frozen peas and 2 tablespoons chopped parsley; simmer 1–2 minutes.
Prepare biscuits: If using canned biscuits, separate and (optionally) cut each biscuit in halves or quarters depending on size — smaller pieces cook more evenly. For homemade biscuits, form into small rounds about 1–1.5 inches thick.
Place biscuits on top: Reduce the heat to low so the stew is at a gentle simmer. Arrange biscuit pieces on top of the simmering chicken mixture — do not press them down or stir in. They need steam to cook.
Cover and steam-cook biscuits: Cover the pot with a tight-fitting lid and let steam-cook the biscuits for 12–16 minutes (time depends on biscuit size and brand). Avoid lifting the lid during this time.
Check for doneness: After 12 minutes, carefully lift the lid and test one biscuit for doneness — the biscuit should be cooked through (no raw dough) and fluffy. If needed, cover and cook 2–4 minutes longer.
Finish & rest: Once biscuits are cooked, remove from heat and let the pot sit, covered, for 3–5 minutes to let the juices settle. Discard the bay leaf. Give a gentle stir around the edges (do not break biscuits apart unless serving).
Serve: Spoon chicken and broth into bowls, making sure each serving gets biscuit pieces. Garnish with extra parsley if desired.