Make a deep vertical cut on each side of the potato and apple.
Attach potato to a spiralizer fitted with the small shredded blade and cut into thin shreds. Repeat with apple. Spread shredded potato and apple on several layers of paper towels; squeeze to release as much moisture as possible. Transfer to a bowl.
Mix egg, flour, maple syrup, salt, and nutmeg into the bowl.
Heat vegetable oil in a skillet over medium heat. Spoon 2 tablespoons of the potato mixture into the skillet; flatten slightly with a spatula. Cook until golden brown, 3 to 4 minutes per side. Drain on a plate lined with paper towels. Repeat with remaining mixture, adding more oil if needed.