1. Make the filling: In a medium saucepan, melt the butter over low heat. As soon as the butter melts, add the flour and increase the heat to medium. Using a wooden spoon and stirring constantly, cook the butter and flour until it is bubbling and just beginning to become slightly golden, about 4 minutes. Using a whisk and whisking constantly, slowly pour in the chicken broth. Keep whisking until the sauce is smooth. Bring to a gentle boil, adjusting the heat as necessary, and cook for 5 minutes. The sauce will thicken slightly to the consistency of a thick syrup. Add the thyme and chicken pieces. Continue cooking, uncovered, to blend the flavors for 15 minutes, then add the chives, salt and pepper to taste. Make the dumpling batter while the chicken cooks in the sauce.

2. Prepare the topping: In a medium bowl, stir the cornmeal, flour, baking powder, and salt together. Using your fingertips, a fork, or a pastry blender, cut the butter pieces until they are the size of peas. You will still see loose flour. In a small bowl and using a fork, stir the milk and egg together. Use a spoon to stir the milk mixture and parsley into the cornmeal mixture, stirring just to evenly moisten the dry ingredients.

3. Drop rounded tablespoons of batter onto the gently bubbling filling. You will have 8 or 9 dumplings. Cover the saucepan and adjust the heat to keep the potpie at a gentle boil, and continue cooking until the dumplings are done, about 10 minutes. Use a knife to cut into one of the dumplings to make sure that they are firm throughout. The sauce will thicken considerably while the dumplings cook.

4. Remove the saucepan from the heat and spoon servings of dumpling and filling onto serving plates.

Variation: Three tablespoons of finely chopped fresh chives can be substituted for the parsley in the dumplings.