Line a rimmed baking sheet with parchment paper and lay on the chicken, skin side up. Drizzle over the oil, and sprinkle evenly with salt and pepper.
Roast until no pink remains in the center, approximately 45-55 minutes.
When the chicken is done, remove any skin or fat and using two forks, shred the meat and set aside.
For the sauce
In a large pot, melt the butter and oil. Add the celery, carrots and garlic and cook on medium low until the vegetables start to soften--about 8 minutes--stirring often so the garlic doesn't burn.
Add the bay leaf and the flour and stir to cook the flour--two minutes. Add the broth and bring to a simmer. Cook until the broth thickens slightly, about 8 to 10 minutes more.
Add the shredded chicken and cream and taste for seasoning. Add salt and pepper as needed. This will depend on a lot on how salty your chicken broth is.
For the dumplings
Once the broth is added and is returning to a simmer as directed above, make the dumplings. Mix the flour, baking powder, and salt in a medium bowl. Stir to combine.
Add the eggs and buttermilk or milk and stir just until no raw flour remains. Do not over mix.
After the broth as simmered for 8 to 10 minutes, then drop the dumplings by tablespoons into the pot. Remember they will puff up, so leave plenty of space between them.
Cover and simmer on low 15 minutes, then ladle up and serve warm.