Heat 2 tablespoons oil over medium-high in a large heavy-bottomed skillet. Add half the turnips, and cook until golden on both sides and tender inside, about 4 minutes per side (turning halfway through). Transfer to a plate, and season both sides with salt and pepper. Repeat with remaining turnips. Drain excess oil.
Return the skillet to low heat, and add the remaining tablespoon oil and the whole sage leaves. Cook, turning leaves until they stop sizzling. Using tongs, transfer the leaves to a paper towel-lined plate, leaving the oil in the skillet.
Increase the heat to medium, and add the shallot. Cook, stirring occasionally, until just starting to brown on the edges, about 3 minutes. Add the cranberries, honey, chopped sage, lemon zest, and cayenne. Season with salt. Cook, stirring, until the cranberries start to burst and the juices are thickened, about 3 minutes.
Serve the turnips topped with the cranberry chutney and sprinkled with the sage leaves.
Recipe by Laura Rege.