Combine first 6 ingredients, 1 teaspoon salt, and 1/4 teaspoon pepper in a Dutch oven; add water to cover. Bring to a boil; cover and simmer 1 1/2 hours. Remove chicken from broth; cool. Strain broth, and set aside.
Remove chicken meat from bone; dice and place in a 10- inch cast-iron skillet. Pour 1/2 cup broth over chicken. Set remaining broth aside.
Combine eggs, milk, 1 cup flour, 1/2 teaspoon salt, and 1/4 cup butter; beat until smooth. Pour over chicken in skillet. Bake at 375° for 1 hour. Cool and cut into wedges.
Cook remaining broth over high heat until reduced to 2 cups. Stir in 1/2 cup flour. Add parsley, 2 tablespoons butter, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Cook over medium heat until thickened and bubbly. Spoon gravy over pudding.