Grate the potatoes, carrot, parsnip and golden beet into a large mixing bowl. Add the chopped green onions and stir to combine. Add the egg and stir again. Sprinkle the flour and salt over the grated vegetables and mix well.
Preheat the oven to 300° F.
In a large, heavy skillet (non-stick or cast iron), heat about 2 Tablespoons of the oil over medium heat. Spoon in about 2 Tablespoons of the potato mixture per pancake, using the back of your spoon to flatten the pancakes out. Cook about 3 minutes or until the pancakes are golden brown on the bottom. Flip them and cook another 3 or 4 minutes, until both sides are golden brown, the edges are crisp, and the pancakes are cooked through.
Put the cooked pancakes on a rack set over a baking sheet. Slide the baking sheet into the warm oven to keep the pancakes warm as you cook the rest in batches. Add more oil to the skillet as needed - you don't want to skimp on the oil or the pancakes will not crisp properly.
Serve the potato pancakes with applesauce or sour cream (or both).