Meat pies and puddings are strictly English in origin. I've known about this recipe all of my adult life - it was one of my grandmother's oldest recipes dating back to the early days when she first got married. It takes a little time but is so well worth it. It is the best chicken dinners I've ever eaten!
Preheat oven to 375 degrees. Combine first 6 ingredients, 1tsp salt and 1/4tsp pepper in a dutch oven; add water to cover. Bring to a boil; cover and simmer 1 1/2 hour. Remove chicken from broth; cool. Strain broth and set aside.
Remove chicken from bone; shred 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, 1/2tsp salt and 1/4 cup butter; beat until smooth. Pour over chicken in skillet.
Bake at 375 degrees for 1 hour. Cool slightly and cut into wedges.
While puddings is baking, cook remaining broth over high heat until reduced to 2 cups. Stir in 1/2 cup flour. Add parsley, 2TBS butter, 1/2tsp salt and 1/4tsp pepper. Cook over medium heat until thickened and bubbly. Spoon gravy over pudding.