In a large Dutch oven or heavy-bottomed pot, melt the butter over medium heat. Add the chopped onion, carrots, and celery. Cook for 5-7 minutes, until softened.

Stir in the minced garlic and cook for 1 minute until fragrant. Sprinkle the flour over the vegetables and stir constantly for 1 minute to cook off the raw flour taste.

Slowly whisk in the chicken broth. Add thyme, salt, and pepper, and bring to a low simmer. Add the chicken breasts, cover, reduce heat to low, and simmer for 15-20 minutes, until the chicken is cooked through.

Remove the chicken and shred it with two forks. Return the shredded chicken to the pot. Stir in the heavy cream and fresh parsley. Keep the stew at a gentle simmer.

In a bowl, whisk together flour, baking powder, sugar, and salt. Cut in the cold butter until the mixture resembles coarse crumbs. Stir in the milk until a soft dough forms. Do not overmix.

Drop spoonfuls of dough onto the simmering stew. Cover tightly and cook on low for 15 minutes without lifting the lid. Dumplings are done when a toothpick inserted into the center comes out clean. Serve immediately.