In a large pot, place the chicken, spices, salt, onions, and carrots, and cover with water. Bring to a boil and simmer for 1-1/2 hours. Strain the broth and reserve. Cut the chicken up into bite-sized pieces. Slice the carrots and reserve.
In a saucepan, heat the butter over moderate heat. Add flour and stir well, forming a roux. Cook for a few minutes, stirring constantly, then add 3 cups of the reserved chicken broth, the wine, salt and pepper. Stir until thickened slightly.
In a small bowl, beat the egg yolks and lemon juice together until frothy, then add to the sauce. Beat well, and continue cooking until sauce thickens more. Remove from the heat.
Preheat the oven to 450 degrees F. In a 4 quart deep-dish baking pan, place the chicken uniformly on the bottom. Arrange the hard-boiled egg slices on top of the chicken, and then top with the ham slices. Spread the reserved carrot slices, then cover with the sauce.
Roll out the pie crust, and lay it over the baking dish. Press along the edge firmly and trim excess. Score in several places to allow steam to escape. Brush dough with egg.
Bake for 10 minutes, then reduce heat to 375 degrees F and bake 30 minutes longer. Crust should be golden brown.