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1
Make crust: Preheat oven to 375u00b0F. 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.
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2
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.
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3
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.
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4
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.
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5
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.
