Set a heavy 9- to 10-inch skillet over high heat with the olive oil and butter. When the butter has melted, stir in the leeks, turnips, and carrots, and cook for 1 minute, stirring once or twice. Season with salt, cover, reduce the heat to medium low, and cook for 5 minutes. Uncover and cook at a medium sizzle, stirring occasionally, until the carrots and turnips are almost completely tender and are nicely -- but lightly -- browned, about 25 minutes. (After about 15 minutes, the vegetables should be just starting to brown, so lower the heat if they're cooking too quickly.)
Reduce the heat to very low and continue cooking until tender, about 5 minutes more. Turn off the heat, season to taste with pepper and more salt, if necessary. Gently fold in the tarragon or basil. Transfer to a warm serving dish, sprinkle with the parsley (or fold it in as well), and serve.