Place oil in a large baking dish. Season turkey tenderloins all over with salt and black pepper and place in baking dish. Arrange potatoes all around turkey and turn to coat with oil. Season potatoes with salt and black pepper. Arrange shallots over potatoes in pan.
In a small bowl, combine wine, broth, vinegar, and tarragon. Pour mixture over turkey.
Roast turkey and potatoes 40 minutes, until an instant-read thermometer registers at least 160 degrees F. Let turkey rest 10 minutes before slicing crosswise into 1/2-inch thick slices. Serve half of the turkey and potatoes with this meal, with all of the broth from the pan over top. Serve spinach on the side. Reserve remaining turkey and potatoes for additional meals.