Prep Time 40 min.
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. Drain through a colander and set aside.
While the potatoes and turnips are boiling, heat oil in a large skillet over medium high heat. Season pork chops with salt and pepper. Once oil is hot, place chops in the skillet and allow to cook for 1 minute. Flip chops, and cook for another minute. Continue to flip the meat about once a minute until meat is at 125° in the middle, about 7-9 minutes. Tip: Frequent flipping actually creates a more even, golden-brown sear without overcooking the surface meat.
Remove pan from heat and add half the sage, garlic, and butter, smashing garlic into butter. Tilt skillet and spoon foaming butter and drippings over pork chops. Transfer pork chops to a cutting board and let rest. Keep the liquid in the pan to use as a gravy
In a small saucepan, melt 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. Return potatoes and turnips to their pot, add the hot milk mixture and mash roughly.Spoon the leftover butter and herbs from the pork over the chops and the mashed turnips. Serve by itself or with a side of green salad.