Season chicken pieces with salt and pepper on cutting board. Melt 3 tablespoons butter in grill or cast iron skillet or large pot over medium-high heat.
Add chicken and cook for 5-6 minutes using tongs, stirring occasionally, until golden brown. Transfer to a plate.
In the same pot, add diced onion, carrots, and celery. Cook for 5 minutes, stirring with wooden spoon, until softened. Add minced garlic and cook for 1 minute until fragrant.
Sprinkle 1/3 cup flour over vegetables and stir constantly for 2 minutes to cook out raw flour taste.
Add bay leaves, thyme, and sage. Return chicken to the pot. Bring to a simmer and cook for 15 minutes, stirring occasionally, until slightly thickened.
While chicken simmers, make dumplings: In a mixing bowl, whisk together 2 cups flour, baking powder, salt, and pepper. Cut in cold butter using a fork or pastry cutter until mixture resembles coarse crumbs.
Stir in milk and chopped parsley just until combined - do not overmix or dumplings will be tough. Dough should be thick and sticky.
Add frozen peas to the pot. Drop dumpling batter by heaping tablespoonfuls onto simmering chicken mixture, spacing them evenly. You should get about 12 dumplings.
Cover pot with a lid and simmer for 15-18 minutes without lifting the lid - steam cooks the dumplings. They're done when a toothpick inserted in the center comes out clean and tops are no longer wet. Remove bay leaves, garnish with fresh parsley, and serve immediately in bowls.