In a large pot, melt 2 tablespoons of unsalted butter over medium heat. Add the chopped onions, diced carrots, and diced celery, stirring frequently for about 5 minutes until the vegetables soften and the onions turn translucent.

Next, add the boneless, skinless chicken thighs to the pot. Season generously with salt and black pepper, then pour in low-sodium chicken broth until the chicken is fully submerged. Bring the mixture to a boil, then lower the heat to a simmer for 30 minutes.

After simmering, carefully remove the chicken thighs from the pot using tongs. Let them cool slightly, then use two forks to shred the chicken into bite-sized pieces. Return the shredded chicken to the pot, along with the frozen peas.

In a mixing bowl, combine all-purpose flour, baking powder, and a pinch of salt. Cut in the remaining 2 tablespoons of butter until the mixture resembles coarse crumbs. Gradually mix in milk and chopped fresh parsley until just combined.

Drop spoonfuls of the dumpling batter into the simmering soup. Cover the pot with a lid and cook for 15 to 20 minutes.

Once the dumplings have cooked, taste the soup and adjust the seasoning with more salt and pepper if necessary. Ladle the chicken and dumplings into bowls and serve hot.