Preheat the oven to 400°F. Grease a 12-inch cast iron skillet with 1 tablespoon of butter.
Working from the outside in, tile sliced beets and turnips in a rosette pattern. I started with red beets on the outer edge and then turnips.
Warm 3 tablespoons of butter in a small skillet set over medium heat. Add onions and cook until soft, stirring frequently (about 4 minutes). Add the garlic and thyme and cook, stirring constantly, for 1 minute more. Take the pan off the heat, and stir in the remaining 5 tablespoons of butter. Once the butter is melted and incorporated, season to taste with kosher salt and freshly ground black pepper.
Pour the butter-garlic mixture evenly over the prepared beets and turnips, then pour over the chicken stock. Cover the skillet tightly with foil, then bake in the oven for 45 minutes. Remove the foil and cook until the top of the gratin is just starting to brown and get crispy (about 30 minutes). Let the gratin cool for 30 minutes. Sprinkle with chopped chive just before serving.