Wash leeks thoroughly to remove any dirt that settles between the layers. Remove the dark green portion of the leeks and coarsely chop the remaining white and light green portions.

Heat oil in a large skillet over medium heat. Add the turnips and leeks and saute until leeks become translucent and turnips are tender.

Transfer leeks and turnips to a food processor or high-power blender (can also use an immersion blender). Blend while adding broth until the mixture gets to your desired texture. Transfer the blended mixture to a saucepan, add salt and tarragon and heat before serving.