Prepare the topping: Put the potatoes and turnips in a large pot and cover with cold water; season with salt. Bring to a boil and cook until tender, about 25 minutes; drain, reserving the pot.
Meanwhile, make the filling: Cook the bacon in a large skillet over medium heat, stirring occasionally, until crisp, about 10 minutes.Drain all but 1 tablespoon of the fat; add the beef to the skillet and cook, stirring occasionally, until browned, about 4 minutes. Add the onion, carrots and celery and cook, stirring, until the vegetables begin to soften, about 4 minutes. Add the mushrooms, thyme and tomato paste and cook, stirring occasionally, until incorporated, about 3 minutes.
Stir the flour into the vegetables until incorporated, about 2 minutes. Add the broth, Worcestershire sauce, 3/4 teaspoon salt and a few grinds of pepper. Cook until slightly thickened, about 3 more minutes. Stir in the parsley.
Preheat the oven to 375 degrees F. Finish the topping: Combine the butter and milk in the reserved pot and cook over medium heat until the butter is melted. Return the potatoes and turnips to the pot and mash with a potato masher or wooden spoon until smooth. Stir in the parmesan and chives and season with salt and pepper.
Transfer the filling to a 2 1/2-quart baking dish and cover with the topping, spreading evenly. Bake until the topping is golden, about 35 minutes. Let rest 15 minutes before serving.
Photograph by Con Poulos
Courtesy of Food Network Magazine