Melt butter in a 6-quart Dutch oven over medium heat. Add onion, carrots and celery and cook for 6 minutes until starting to soften.
Add chicken, thyme, pepper and 1/2 teaspoon salt and cook for 4 minutes, stirring. Sprinkle with the 3 tablespoons gravy flour and stir for 1 minute.
Gradually stir in broth and the 1/2 cup gravy milk. Bring to a gentle simmer and cook for 8 minutes, stirring occasionally, until the chicken reaches 165°F; fold in peas.
Whisk dumpling flour, cornmeal, baking powder and remaining 1/4 teaspoon salt in a bowl. Stir in the 1/2 cup dumpling milk and oil just until a soft dough forms.
Drop dough in 10 equal mounds over the gently simmering gravy, leaving gaps between them. Cover tightly and cook over low heat for 15 minutes without lifting the lid.
Check one dumpling by splitting its center; cook covered for 2 to 3 minutes more if any wet dough remains. Serve 2 dumplings with one-fifth of the chicken and gravy per portion.