Pat the chicken pieces dry and season with 3/4 teaspoon of the salt and the black pepper. Heat 1 tablespoon of the olive oil in the large deep skillet over medium-high heat until shimmering. Sear the chicken in a single layer, turning once, until golden on the outside, about 4–5 minutes. It does not need to be cooked through — it will finish in the stew. Transfer to a plate.

Add the remaining 1 tablespoon olive oil to the same skillet. Add the onion, carrots, and celery with the remaining 1/4 teaspoon salt; cook, stirring, until the onion is translucent and the carrots are just starting to soften, about 3 minutes. Stir in the garlic and cook 1 minute more, until fragrant.

Sprinkle the flour over the vegetables and stir constantly for 1 minute, until the flour coats everything and smells toasty. Slowly pour in the chicken broth, whisking as you go to avoid lumps. Stir in the thyme and bay leaf, and bring to a steady simmer — the broth should thicken into a light gravy within 2–3 minutes.

While the broth comes to a simmer, separate the biscuits and cut each one into 6 to 8 pieces. Return the chicken and any juices to the skillet and stir it in. Arrange the biscuit pieces in a single layer over the top of the stew, leaving a little space between them so they have room to puff.

Reduce the heat to low, cover with the lid, and cook until the dumplings are puffed, set, and golden-topped, 6–8 minutes. A toothpick inserted into the thickest dumpling should come out clean with no wet dough. The chicken finishes cooking through in the simmering stew during this time.

Uncover and fish out the bay leaf. Gently stir in the frozen peas and the half-and-half, being careful not to break the dumplings apart. Simmer 1–2 minutes, just until the peas are heated through and the sauce is creamy. Shower with parsley and serve straight from the skillet.