Peel the potatoes, parsnips, celery root and turnips and cut into 1-to-2-inch pieces. Combine in a large pot and cover with twice the volume of cold water; add 1 tablespoon salt. Bring the water to a boil over high heat, reduce the heat to medium and boil gently until the vegetables are tender, 20 to 30 minutes. (Insert a paring knife into a potato; there should be no resistance.)
Drain the vegetables well and return to the pot. Let the pot sit 5 minutes over low heat to cook off any excess liquid. Run the hot vegetables through a food mill into a large bowl. Add 1 1/2 teaspoons salt, then whip in the butter.