Pat the chicken dry and season with 1/2 teaspoon salt and the black pepper. Heat the olive oil in a large Dutch oven over medium-high heat. Brown the chicken in batches for 3 to 4 minutes, then transfer it to a plate. It does not need to be fully cooked yet.
Reduce the heat to medium and add the butter. Add the onion, carrots, and celery and cook for 6 to 8 minutes, until beginning to soften. Stir in the garlic, thyme, and sage and cook for 30 seconds.
Sprinkle 1/4 cup flour over the vegetables and stir for 1 minute. Slowly pour in the chicken broth while scraping the pot. Stir in 1/2 cup pumpkin puree, then return the chicken and its juices to the pot.
Bring to a gentle simmer and cook uncovered for 10 minutes. Stir in the heavy cream. Keep the stew at a low simmer while preparing the dumplings.
Whisk 2 cups flour, baking powder, 1/2 teaspoon salt, and chopped sage in a medium bowl.
Stir together 1/2 cup pumpkin puree, buttermilk, and melted butter. Fold the wet mixture into the dry ingredients just until combined. The dough should be thick, soft, and scoopable.
Use a medium cookie scoop or two spoons to drop 10 to 12 portions of dough over the simmering stew, leaving a little space between them.
Reduce the heat to low, cover the Dutch oven tightly, and cook for 15 minutes without lifting the lid. Check a center dumpling; if the inside is still wet, cover and cook for 2 to 3 minutes more.
Confirm that the chicken has reached 165°F. Taste the broth and add the remaining salt, more pepper, or lemon juice as needed. Serve hot with fresh thyme or sage.