Preheat oven to 180°C. Place oven rack in lowest position.
In large roasting pan or 2 smaller pans, combine potatoes, parsnip, celeriac, onion, thyme, salt, pepper and oil. Place spatchcock, breast-side up, in among the vegetables; sprinkle with paprika.
Bake for 50 minutes, stirring vegetables occasionally. Stir honey into the vegetables. Bake until spatchcock are cooked through and vegetables are fork-tender, 10 to 15 minutes. Cut each spatchcock in half to serve.
To cut the amount of saturated fat in these dishes by half, remove the skin from the chicken pieces and spatchcock after cooking and discard. To remove more fat from the stew, refrigerate it overnight and then scoop off all the solidified fat the next day. Reheat the stew and serve.