Preheat your oven: Set your oven to 375°F (190°C). Grease or line a 12-cup muffin tin with paper liners to prepare for baking.
Prepare the vegetables: Take the grated zucchini and potatoes and place them in a clean kitchen towel or cheesecloth. Squeeze out as much moisture as possible to prevent the muffins from becoming soggy.
Mix the dry ingredients: In a large mixing bowl, combine the grated zucchini, grated potatoes, chopped onion, minced garlic, all-purpose flour, and grated Parmesan cheese. Stir the mixture to evenly distribute all dry ingredients.
Add the wet ingredients: Next, add the baking powder and the lightly beaten eggs to the mixture. Season with salt and pepper to taste, adjusting to your preference.
Incorporate the oil: Pour in the 1/4 cup of vegetable oil and stir until the mixture is well combined. Ensure that the vegetables are coated uniformly.
Fill the muffin cups: Divide the mixture evenly among the prepared muffin cups, filling each about 3/4 full. This will allow space for the muffins to rise while baking.
Bake the muffins: Place the muffin tin in the preheated oven and bake for 20-25 minutes. The muffins should turn golden brown, and you can insert a toothpick into the center; it should come out clean when they are done.
Cool and serve: Once baked, allow the muffins to cool in the tin for a few minutes before carefully transferring them to a wire rack to cool completely. Serve warm, topped with a dollop of sour cream and a sprinkle of chopped chives for a delightful presentation.