Preheat the oven to 200°F. Place a wire cooling rack on a large baking sheet.
Place the grated zucchini, potatoes, and onion in a colander or fine mesh sieve over a bowl and allow the liquid to drain for 10 minutes. Note: Your potatoes may turn pinkish-brown. This is normal and is just a reaction of an enzyme in the potatoes with the air which won't affect the flavor of your pancakes.
Using the back of a spatula or large spoon, press down on the zucchini, potatoes, and onions and squeeze out any remaining liquid. Note: It's very important to remove as much liquid as possible. If your pancake batter is too wet, they will not hold together well.
Place the zucchini, potatoes, and onion in a large bowl. Add the egg, flour, and salt and pepper to taste (I usually add 1 teaspoon salt and ½ teaspoon pepper). Mix until well combined. If the mixture seems way too wet, add a little more flour.
Heat the oil in a large frying pan over medium to medium-high heat until hot (I use a 12-inch cast iron skillet, and the 1 cup of oil is just the right amount for this size pan). To test the oil, you can drop a small amount of the batter into the oil. If it immediately sizzles and bubbles vigorously, the oil is hot enough.
Using a ¼-cup measure for each pancake, carefully drop the batter into the hot oil. Immediately flatten out the pancakes with the back of a fork (they should be about ¼-inch thick; you don't want them too thick or else they will get too brown before they cook through). I can fit 3 pancakes in my 12-inch cast-iron skillet.
Fry the pancakes until the bottoms are golden brown and crispy, about 2 minutes. Watch the heat and adjust up or down as needed. Carefully flip the pancakes and cook on the other side until golden brown and crispy, about 2 more minutes.
Remove the pancakes from the oil and place on a wire rack placed on a baking sheet. Immediately sprinkle with salt to taste. Place the cooked pancakes in the preheated 200°F oven to stay warm and crispy while you cook the remaining pancakes.
These are best when served immediately! Serve with sour cream or applesauce (or both) for dipping.