Sprinkle chicken with salt and pepper. Heat a 6-quart Dutch oven over medium heat; add thighs. Cook, turning once, until nicely browned on both sides and cooked through, 8 to 10 minutes. Repeat with breasts; cut breasts crosswise into thirds.
Add butter to pot; when melted, add onion, cover, and cook, stirring occasionally, until softened, about 6 minutes. Stir in broth and thyme; bring to a boil. Combine cornstarch and 2 tablespoons flour in a bowl. Gradually whisk in milk, and whisk milk mixture into broth. Boil, whisking, until thickened, about 5 minutes. Return chicken to Dutch oven; remove from heat while you prepare the vegetables.
Bring a saucepan of water to a boil; add salt. Add carrots and green beans; cook until crisp-tender, about 5 minutes. Drain in a colander, then rinse with cold water to stop cooking. Stir into chicken mixture.
Prepare dumpling dough: In a medium bowl, whisk together flour, cornmeal, baking powder, and salt. Add butter; cut into dry mixture with fingers until mealy. Pour in milk; stir with a fork until dough comes together. Return chicken mixture to a simmer, and drop dough by tablespoons on surface of stew. Cover; cook for 15 minutes without lifting lid. (The dumplings will puff up.) Serve.