First, grate the turnips and place them in a colander in the sink. Let them sit while you prepare the rest of the ingredients.
In a medium saucepan over medium heat, combine the apples, apple cider vinegar, maple syrup, golden raisins, cinnamon, and nutmeg. Bring to a gentle boil, then reduce heat to medium-low and simmer uncovered for the time it takes to prep and cook the turnip cakes (20-25 mins). Be sure to stir the chutney occasionally to prevent scorching. You'll know it's finished when the apples and raisins are tender and the liquid has evaporated.
Come back to the turnips, and use your hands to squeeze any excess water out of them over the colander. Do this a few times to try to get out as much water as possible. Pat them dry with a paper towel and transfer them in a mixing bowl. Add the scallions, garlic, cornmeal, salt, pepper, and beaten egg. Mix until the batter comes together. Take one to two heaping tablespoons of the batter in your hands and form it into a cake that's ½-inch thick and ~2 inches wide. Repeat with the rest of the batter until you have 8 cakes.
In a large skillet, heat the olive oil over medium heat until hot. Place a few cakes into the skillet, leaving enough room between them to flip them with a spatula (I usually can fit 4 in a large skillet). Cook for 2 to 3 minutes on each side until golden brown. Repeat with the rest of the cakes.
Serve them warm with apple chutney!