If you'd like to keep the fritters warm between batches, preheat your oven to 200 degrees F. Spread the shredded zucchini, potatoes, and onion onto a clean kitchen towel and press out as much moisture as possible, changing the towel once or twice as needed.
Place the shredded vegetables in a large bowl, then add the eggs, breadcrumbs, flour, garlic powder, salt, pepper, and baking soda. Mix until completely combined, using your fingers if needed to make sure all of the ingredients are evenly distributed. Fold in the shredded cheese.
In a large nonstick skillet, heat 1 tablespoon of the olive oil over medium heat. Scoop a scant 1/4-cup portion of the fritter batter onto the frying pan and flatten into an even layer, using either the back of the measuring spoon or the bottom of a drinking glass. Cook for 3 to 4 minutes until golden on the first side, then flip and continue cooking until the second side is golden as well. Depending upon your skillet, you may need to adjust the heat as you go to ensure the pancakes cook all the way through but still get crisp on the outside. Repeat with the remaining pancakes, keeping batches warm in the oven as desired. If excess liquid collects in the bottom of the bowl, discard it. Serve warm, topped with Greek yogurt and fresh chives.
Be sure to measure the shredded zucchini and potatoes. If you use a different amount than called for in the recipe, the fritters may not hold together.
I have not tried this recipe gluten free, but I believe you could swap oat flour for the whole wheat flour and gluten free bread crumbs, plus a pinch of extra salt and pepper for the seasoned breadcrumbs.
Store leftovers in the refrigerator for up to three days. Reheat gently on the stovetop with a bit of olive oil to prevent sticking.