Add the turnips and potatoes to a large pot of cold salted water. Bring to a boil, then reduce to a simmer and cook until soft and tender, about 20 minutes.
In a small saucepan, add the butter over medium-high heat. Once melted and foamy, add the shallots and sage and saute until the shallot is tender, the sage is fragrant and the butter begins to brown. Season with salt and pepper. Add the milk and bring to a simmer. (This will infuse the milk with the sage flavor.)
Drain the turnips and potatoes in a colander in the sink. Add the turnips and potatoes back to the pot they were cooked in. Add the hot milk and roughly mash the vegetables. If you prefer a smoother texture, use an immersion blender and puree until smooth and silky. Taste for seasoning, add salt and pepper.