Gather the ingredients.
In a large Dutch oven or stockpot, combine chicken, water, parsley, celery, half of the onions, and thyme. Cover and cook chicken on the stovetop over medium heat until very tender, about 1 1/4 hours.
Drain the chicken, reserving stock. Cool chicken. Skin, bone, and chop meat. Strain stock.
Spread corn and green beans in a buttered 9 x 13-inch baking pan. Arrange chicken over the vegetables.
Combine cornstarch and reserved stock in a medium saucepan and stir until dissolved. Bring to a boil, stirring constantly. Pour sauce over chicken.
Preheat oven to 325 F.
Melt butter in a small saucepan. Add remaining 1/2 cup of onions and cook over low heat until soft.
In a medium bowl, toss the bread crumbs with the butter and sauteed onions, the beaten egg, sage, and pepper.
Sprinkle bread crumb mixture over chicken and bake for 25 minutes, or until sauce is bubbly and the top is browned.