Make crust: Preheat oven to 375°F. Mix flour, salt and marjoram in a medium bowl. Add shortening and blend with a pastry cutter until mixture resembles coarse meal. Add milk and stir lightly; add extra if dough is too dry. Shape dough into a rectangle, wrap in plastic and refrigerate.
Make filling: Melt 1 Tbsp. butter in a large Dutch oven over medium heat. Add onions; cook until lightly browned, 5 to 7 minutes. Transfer to a bowl. Season chicken with salt and pepper. Melt remaining butter in Dutch oven. Cook thigh pieces until browned; add broth and cook, covered, over low heat for 15 minutes.
Add chicken breast pieces, celery, mushrooms, ham, onions and marjoram to Dutch oven. Cook, covered, until breast pieces are cooked through, 10 minutes. Mix flour, 1/2 cup water, and Worcestershire and browning sauces. Whisk into chicken mixture and cook, stirring, until thickened. Season with salt. Transfer to a shallow 2-quart baking dish.
Roll out dough between floured sheets of waxed paper to fit baking dish. Roll crust onto a rolling pin and unroll over filling. Press edges with a fork to crimp. Slice 1-inch slits in crust to vent steam. Brush crust with egg mixture.
Place baking dish on a larger, foil-lined, rimmed sheet pan; bake pie until crust is golden and filling bubbles, 35 to 40 minutes. Serve immediately.