Trim leaves and most of stems from the turnips and place in a bowl of water. Swish around to remove any sand or dirt from the leaves. Transfer the greens to a salad spinner and dry them. Set aside.
Rinse turnips (check the stems where sand collects) and cut into quarters. Slice potatoes into quarters about the same size as the turnips.
Fill a medium pot with water and kosher salt and bring to a boil. Add potatoes and turnips, Reduce heat to a rapid simmer and cook vegetables until tender, about 15 minutes. Drain.
In a medium skillet or cast iron pan, heat 1/2 teaspoon butter and 2 teaspoons olive oil over medium to medium high heat. Add sliced turnips and potatoes. Saute vegetables until crispy and browned. Transfer to a bowl.
Add remaining butter and olive oil to the pan. Add shallots and sauté for 2-3 minutes until slightly tender. Add the greens and cook until just wilted. Add the potatoes back to the pan and toss to combine.