Butter a 9-by-13-inch baking dish.
In a saute pan over medium heat, add 1 tablespoon each of the butter and oil. When the butter is melted, add the celery, carrots, onion and fennel (mirepoix) and saute until caramelized, about 10 minutes. Add the garlic and stir through. Scoop out 2 cups of the caramelized mirepoix and reserve for the gravy. Put the remaining caramelized mirepoix in a large bowl (you should have about 4 cups).
Heat a second saute pan over medium heat. Add the remaining 1 tablespoon each butter and oil and heat until the butter is melted. Add the celery root, parsnips and turnips and saute until caramelized, about 10 minutes. Transfer the mixture to the bowl with the mirepoix and add the stuffing mix. Add the chicken broth, parsley, salt and pepper and toss together. Transfer the stuffing mixture to the prepared baking dish.
If baking right away, preheat the oven to 350 degrees F. Cover with foil and bake until the stuffing is heated through, about 1 hour.
If making ahead of time, cover the unbaked stuffing with foil and refrigerate overnight. The next day, preheat the oven to 350 degrees F. Bring the stuffing to room temperature, then drizzle with 1/2 cup of chicken broth. Bake, covered with foil, until heated through, about 1 hour.