Rich Stock: Combine all ingredients in a pot; add water to cover if needed, and bring to a boil; reduce heat and simmer, uncovered, for 1 hour.
Savory Dough: In a food processor, combine flour, salt and thyme; add butter and pulse until mixture resembles corn meal; with machine running, add ice water and egg yolk; process just until dough holds together.
Place dough on plastic wrap; flatten into a disk and wrap well; refrigerate at least 1 hour.
Filling: Remove chicken from stock; cool, discard skin and bones, and shred or cut chicken into bite-sized pieces; strain stock, discarding solids and set aside 2 cups; refrigerate or freeze remaining stock for another use.
Preheat oven to 375°; fry bacon in a large skillet until crisp; remove and drain on paper towels, leaving 2 T drippings in skillet; set bacon aside.
In the same skillet, over medium-high heat, saute onions and potatoes until potatoes begin to turn golden, 4-5 minutes; add leeks, carrots, and mushrooms and saute another 4-5 minutes.
Knead butter and flour together; add to skillet and cook, stirring, until melted and smooth; stir in wine, stock and milk and simmer, stirring constantly, until thick and bubbly, 2-3 minutes.
Add hot pepper sauce, Worcestershire, thyme, parsley, salt, pepper and lemon zest; taste and adjust seasoning; stir in chicken, bacon and hard-boiled eggs; pour into a 9" x 9" baking dish or other casserole; set aside.
Roll out dough to 1/4" thickness and transfer to a baking sheet; chill 15 minutes; whisk egg yolk and cream together; working quickly, place dough over top of filling and cut 1" larger than rim of dish; tuck extra 1" of dough inside around the edges.
Vent, brush with egg wash and bake on baking sheet 35-40 minutes, or until crust is golden.