In a 12-inch skillet, heat 1/4 cup of the oil over medium-high heat. Add the potato, celery root, onion, turnip, rutabaga, garlic and thyme, and cook, stirring occasionally, until the vegetables are golden-brown on all sides, 7 to 8 minutes.
Add 3/4 cup of the brisket cooking liquid. Cook on medium-high heat to reduce the liquid to a syrupy consistency, about 2 minutes. Add 2 Tbs. of the butter and stir until melted and well mixed with the vegetables. Remove from the heat and set aside.
Heat the remaining 2 Tbs. oil in a large (11- to 12-inch) cast iron pan or heavy-duty skillet over high heat. Add the corned beef in a single layer. Sear without stirring until it develops good color, about 3 minutes. Stir the beef and add 1 Tbs. of additional oil if the pan looks dry. Sear for about 3 more minutes to brown on the other sides.
Fold in the vegetables and the remaining 1/4 cup cooking liquid. Stir in the remaining 1 Tbs. of butter, and the parsley and chives. Cook over medium heat until the liquid is completely evaporated, about 5 minutes. Season to taste with salt and pepper.
The hash is delicious topped with a soft poached egg and hollandaise sauce.