Season the chicken all over with salt and pepper. Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Sear the chicken for 3 to 4 minutes per side until golden brown. Transfer to a plate and set aside.
Reduce the heat to medium. Add the butter to the same pot. Once melted, add the onion, carrots, and celery and cook, stirring occasionally, for 4 to 5 minutes until softened.
Add the garlic and cook, stirring constantly, for 1 minute more until fragrant.
Sprinkle the flour and poultry seasoning over the vegetables and stir to coat. Cook for 1 to 2 minutes, stirring constantly, until lightly golden.
Slowly pour in the chicken broth and water (or more broth), stirring as you go to prevent lumps.
Nestle the seared chicken back into the pot. Bring to a boil over medium-high heat, then reduce the heat to medium-low. Cover and simmer for 20 to 25 minutes until the chicken is cooked through and very tender.
Transfer the chicken to a cutting board and shred into bite-sized pieces using tongs or two forks. Return the shredded chicken to the pot and stir to combine. Taste and adjust seasoning with salt and pepper as needed. Keep the soup at a gentle simmer over medium-low heat.
In a medium bowl, whisk together the flour, baking powder, salt, and garlic powder. Add the melted butter and milk and stir until just combined. Do not overmix. The dough will be thick and slightly sticky.
Using a tablespoon cookie scoop or two spoons, drop the dumpling dough directly into the gently simmering soup, spacing them slightly apart. Cover the pot with a lid and cook for 15 to 18 minutes without lifting the lid, until the dumplings are fluffy and cooked through. To test, insert a toothpick into the center of a dumpling. It should come out clean.
Pour in the amount of heavy cream to reach your desired thickness and simmer 1 minute. Taste and adjust seasonings as needed.
Ladle into bowls and garnish with a sprinkle of fresh parsley or chives if desired. Serve immediately.