Step 1

In a Dutch oven (or a 5- to 6-quart heavy pot with a tight-fitting lid), heat butter over medium. Add onion, carrots, and thyme. Cover and cook, stirring occasionally, until onion is soft, about 5 minutes.

Step 2

Add 1/4 cup flour and cook, stirring, 30 seconds. Add broth and bring to a boil, stirring constantly; season with salt and pepper. Nestle chicken in pot; reduce heat to medium-low. Cover and cook, stirring occasionally, 20 minutes.

Step 3

Meanwhile, make dumplings: In a medium bowl, whisk together remaining 3/4 cup flour, dill, baking powder, and 1/2 teaspoon salt. With a fork, gradually stir in 1/2 cup milk to form a moist and soft batter. It should be just a little thicker than pancake batter and should easily drop from the tip of a spoon. (Add additional 2 tablespoons milk if too thick.) Set aside.

Step 4

Stir peas into pot. Drop batter in simmering liquid in 10 heaping tablespoonfuls, keeping them spaced apart (dumplings will swell as they cook). Cover, and simmer until chicken is tender and dumplings are firm, 20 minutes. Serve.