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

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

Meanwhile, make the dumplings: In a medium sized bowl, whisk together the remaining 3/4 cup flour, chopped 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; you can add an additional 2 tablespoons of milk if too thick. Set aside.

Pour the peas into the pot. And using a cookie scooper or spoon, drop the batter in the simmering liquid in 10 heaping tablespoonfuls (or smaller if desired), keeping them spaced apart. The dumplings will swell as they cook. Flip the dumplings to ensure even cooking. Cover and simmer until chicken is cooked through and tender and the dumplings are firm, for about 20-25 minutes. Serve.