Melt the dairy free butter in a large Dutch oven over medium heat. Add the onion, carrots, and celery and cook for 6 minutes until softened, then stir in the garlic for 30 seconds.
Stir in the flour and cook for 1 minute, then slowly whisk in the chicken broth until smooth. Add the whole chicken thighs, thyme, bay leaf, salt, and black pepper, then cover and simmer for 20-25 minutes until the chicken is tender.
While the stew simmers, whisk the flour, baking powder, sugar, and salt in a bowl. Cut in the cold dairy free butter until it looks like coarse crumbs, then stir in the almond milk and half the parsley just until a thick, sticky dough forms.
Remove the chicken, shred it with two forks, and return it to the pot along with the oat milk. Discard the bay leaf and bring the stew back to a gentle simmer.
Drop heaping tablespoons of dough over the surface of the stew. Cover tightly and simmer for 15 minutes without lifting the lid, until the dumplings are puffed and cooked through.
Let the pot rest for 5 minutes, then sprinkle with the remaining parsley and serve.