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