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.