In a large cast iron skillet, heat the olive oil over a high flame.

Carefully add the turnips and potatoes to pan. Cook, stirring occasionally, until tender and dark browned on all sides, about 30 minutes. Add the onions, season with the paprika and salt. Turn the heat down to medium-low and saute for about 5 minutes, until translucent and sweet.

To serve, spoon the hash onto plates, drizzle with the vinaigrette and top with coarsely torn basil leaves.