Peel the turnips and put them in a large bowl with the sugar and pinch of fine salt. Toss to coat the turnips thoroughly.
In a large pot, melt the butter. When it is blond and foamy, add the turnips to the pot in a single layer and brown on all sides. When they are golden brown, add a third of the chicken broth and cook, covered, over low heat until the broth has been completely absorbed. Repeat a second and then a third time: When this is done, the turnips are cooked. (Turnips should never be left dry in the pot. If the broth reduces too quickly, add a bit of water when necessary.)
Drizzle with a few spoons of chicken jus or olive oil and bring to a simmer. When it begins to bubble, sprinkle with parsley and serve immediately.
