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 sauté 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.

Drain the turnips and potatoes in a colander. Add the turnips and potatoes back to the pot they were cooked in. Pour in the hot milk and roughly mash the vegetables. We like this dish a bit lumpy. Taste for seasoning and add salt and pepper if desired.