To make the filling, heat the oil and butter in a large skillet set over medium-high heat. When the foam subsides, add the mushrooms, onions and thyme or rosemary; sauté until mushrooms brown, about 8 minutes. Add the sausage, squeezed out of their casings, and sauté until no longer pink, breaking up with spoon. Add turkey or chicken, season with salt and pepper and if it's uncooked, sauté until chicken is opaque, about 5 minutes. Add Madeira or wine and cook, scraping up the browned bits on the bottom of the pan, for about 2 minutes.
Mix the flour with about a tablespoon of butter or oil and add it to the pan along with the broth; bring to a boil. (Alternatively: shake the flour over the meat, toss it around to coat, then pour in the broth.) Simmer until sauce thickens, stirring often, about 3 minutes. Stir in the parsley and season with salt and pepper. Transfer to a baking dish and top with egg slices, if you like. (The pie can be made ahead to this point; cover and chill overnight and when it's time to bake, cover with pastry first.)
When you're ready to bake the pie, preheat the oven to 400°F. Roll out the puff pastry or regular pastry dough on a lightly floured surface until it's about an inch bigger around than your baking dish; drape over the filling. You could leave it over the edge of the baking dish, or crimp it. Cut a few slits in the top and if you like, brush with some lightly beaten egg. Bake pie for 45 minutes, or until bubbly and golden.
Let rest 15 minutes before serving.