Melt butter in a large stockpot or Dutch oven over medium heat. Add onion, carrots and celery. Cook, stirring occasionally, until tender, about 3-4 minutes.
Stir in garlic and poultry seasoning until fragrant, about 1 minute.
Whisk in flour until lightly browned, about 1 minute.
Stir in chicken, chicken stock and wine. Bring to a boil; reduce heat and simmer, stirring occasionally, until thickened and flavors have blended, about 15 minutes.
Stir in heavy cream; season with salt and pepper, to taste.
Working in batches, gently stir in dumplings. Cover, reduce heat and simmer, stirring occasionally, until dumplings are cooked through and tender, about 5-8 minutes.
Stir in peas and parsley until heated through, about 1-2 minutes.
Serve immediately.
for the dumplings
Grate butter using the large holes of a box grater. Stir into the flour.
Add buttermilk and stir using a rubber spatula until a soft dough forms.
Working on a lightly floured surface, knead the dough 3-4 times until it comes together. Using a rolling pin, roll the dough into 1/4-inch thickness. Cut dough into bite-size pieces.