Pat the chicken thighs dry and season both sides with the kosher salt and black pepper.
Heat the neutral oil in a large Dutch oven over medium-high heat. Add the chicken thighs skin side down and cook until well browned, about 5 to 7 minutes. Turn and brown the second side, about 3 to 4 minutes. Transfer the chicken to a plate.
Drain off the excess fat, leaving about 2 tablespoons in the pot. Reduce the heat to medium. Add the onion, celery, and carrots and cook, stirring occasionally, until softened, about 5 to 6 minutes.
Add the garlic and cook 30 seconds. Add the flour, stir, and cook 1 minute.
Pour in the chicken broth, scraping up the browned bits from the bottom of the pot. Add the poultry seasoning and dried thyme. Return the chicken thighs and any accumulated juices to the pot.
Bring the broth to a gentle simmer. Cover and cook until the chicken is tender, about 35 to 45 minutes. Chicken is safe at an internal temperature of 165°F (74°C).
Transfer the chicken thighs to a plate. When cool enough to handle, remove and discard the skin and bones. Shred or chop the chicken into bite-size pieces.
Return the shredded chicken to the pot, along with the heavy cream, and bring the broth back to a gentle simmer. Taste and season with additional salt and black pepper as needed.
Dumplings
In a medium bowl, whisk together the flour, baking powder, and salt. Cut in the cold butter with a pastry cutter, fork, or fingertips until the mixture resembles coarse crumbs with some pea-size pieces of butter. Stir in the heavy cream just until a thick dough forms. If the dough appears dry, add more cream 1 teaspoon at a time.
Break the dough into rounds, about 10-12 golf-ball sized rounds, and drop them into the gently simmering chicken mixture, leaving space between each dumpling.
Cover the pot and cook without stirring for 12 to 15 minutes, until the dumplings are puffed and cooked through.
Remove the pot from the heat and let stand, covered, for 5 minutes. Serve hot.
Notes
Chicken: Bone-in, skin-on thighs give the broth the most flavor. Boneless or skinless can be used, but brown them anyway and expect a lighter broth. You could use chicken breast if that's what you prefer.
Broth: Use low-sodium chicken broth to control the amount of salt in the final dish.
Shortcut: Cooked rotisserie or leftover chicken can be used. Skip the browning step, cook the vegetables in butter or oil, and stir the shredded chicken into the broth before adding the dumplings.
Lid: Do not lift the lid or stir while the dumplings cook. They cook by trapped steam.
Storage: Refrigerate up to 3 days. Reheat gently, covered, with a splash of broth. Freezing with the dumplings in is not recommended.