Heat a large saucepan over medium-high heat. Add the sausage and 1 tablespoon olive oil. Cook until the sausage starts to brown. Pour in the water, cover, and bring it up to a boil. Cook for 15 minutes.
Meanwhile, wash the roots and greens thoroughly. If you’re using Purple Top or another variety of large turnip, separate the roots from the greens and cut the roots into quarters. If you’re using Hakureis, you can leave them whole.
Make sure you still have 2 quarts of water in the saucepan and add the roots and greens at the same time. They will not be completely submerged right away, but check back in a minute or two and stir things around. The greens will have cooked down, and both the roots and greens should be submerged in the cooking liquid. If they are not, add just enough water to barely cover them.
Simmer covered for about 10-15 minutes. You’re looking for the roots to be quite tender and the greens to be soft. Once they’re done, drain off all but 1/2 cup — pot liquour is the life-fixing chicken soup of the South.
Add the salt. If you have a collard chopper, roughly work through the roots, greens and sausage. The end product ain’t gonna be pretty, but it is delicious. If you don’t have a collard chopper, get one, and use a large fork in the meantime. Serve warm.