1 (about 6 pounds) roasting hen or chicken
1 medium onion, thinly sliced
4 teaspoons salt, or to taste
2 teaspoons pepper, or to taste
1/2 stick (2 ounces) unsalted butter
1/4 cup flour, or more as needed
3 cups milk, whole or 2 percent
1 batch Bisquick dumplings, with a few extra tablespoons of milk added to make the dough soft enough to spoon
In a large pot, combine the chicken, onion, salt and pepper. Add enough water (about 8 cups) just to cover the chicken. Bring the water to a simmer and let the chicken cook until tender, about 1 1/2 hours.
When the chicken is tender, remove it from the cooking liquid. Strain the liquid and set it aside. Discard the onions. Carefully remove the chicken meat from the bones. Discard the skin and bones. Cut or tear the chicken meat into bite-size pieces. Set aside.
In a large pot, melt the butter. Add the flour and cook, stirring constantly for a minute. Pour in 4 cups of the reserved cooking liquid, whisking to combine. Add the milk and allow the sauce to come to a boil, then reduce the heat. The sauce should be only slightly thickened, more souplike than saucelike. Add the chicken pieces and taste, adjusting seasonings as desired. If the sauce is thick, add more of the reserved chicken stock.
Bring the mixture to a simmer, add golf-ball-size spoonfuls of the dumpling dough, dividing the dough so there are 12 dumplings. Adjust the heat so the sauce just simmers, with the cover off the pot. Cook the dumplings for 10 minutes. Cover and let cook for 10 more minutes. Serve immediately.