Brown the chicken. Pat the chicken dry and season with salt and pepper.
Heat olive oil in a large pot over medium-high. Sear the chicken skin-side down until golden, 4–5 minutes per side. Remove to a plate.
Build the base. Reduce heat to medium.
Add onion, celery, and carrots with a pinch of salt. Cook, stirring, until softened, 6–8 minutes. Stir in garlic and thyme; cook 30 seconds until fragrant.
Make a quick roux. Add 2 tbsp butter to the pot.
When melted, sprinkle in 1/4 cup flour and stir for 1 minute to coat the veggies. This will slightly thicken the broth later.
Add broth and simmer. Pour in the chicken broth while stirring. Add bay leaf and return the chicken (and any juices) to the pot.
Bring to a gentle simmer, cover, and cook for 25–30 minutes, until the chicken is cooked through and tender.
Shred the chicken. Transfer the chicken to a board. Discard skin and bones; shred meat into bite-size pieces. Skim any excess fat from the pot if you like.
Return chicken to the pot.
Optional richness. Stir in milk or cream if using. Taste and adjust seasoning with salt and pepper. Keep the stew at a steady simmer.
Make the dumpling dough. In a bowl, whisk 2 cups flour, baking powder, and 1 tsp salt.
Cut in 2 tbsp cold butter with fingers or a fork until crumbly. Stir in 3/4 cup buttermilk just until a shaggy dough forms; add a splash more if dry. Do not overmix.
Drop the dumplings. Using a spoon or small scoop, drop tablespoon-sized mounds of dough onto the simmering surface.
Leave space between them—they’ll puff. Cover the pot and cook undisturbed for 12–15 minutes, until dumplings are cooked through.
Finish and brighten. Gently stir in peas and parsley. Cook 2 minutes more.
Taste again for salt and pepper.
Serve. Ladle into warm bowls. The dumplings should be tender and the broth creamy but not heavy.