Cut the celeriac and potatoes into roughly equal-sized pieces, 1 or 2 inches in diameter. Place in a pot with water to cover; add a handful of salt. Bring to a boil and cook until both potatoes and celeriac are tender, about 15 minutes.
Drain the vegetables well and rinse out the pot. Put the vegetables through a food mill placed over the pot, or mash them with a large fork or potato masher. Add the butter and, gradually, the milk, beating with a wooden spoon. When the mixture is smooth, season it with salt and pepper and serve, keep warm, or allow to cool for reheating later. Garnish before serving.