Preheat your oven to 375°F (190°C). Grease a standard 12-cup muffin tin generously with non-stick spray or line with parchment paper liners.
Grate the zucchini and peeled potato using the coarse side of a box grater.
Place the grated zucchini in a colander, toss with 1/2 teaspoon of salt, and let sit for 10 minutes to release water.
Transfer grated zucchini and grated potato into the center of a clean, lint-free kitchen towel or cheesecloth. Wrap tightly and twist, squeezing forcefully over the sink until no more liquid drains out. The vegetables should feel compact and dry.
In a large mixing bowl, whisk together the eggs, olive oil (or melted butter), milk, garlic powder, onion powder, salt, and black pepper until well blended.
In a separate bowl, whisk together flour, baking powder, and baking soda.
Add the squeezed zucchini, potato, chopped green onions, and 1 1/4 cups of shredded cheddar cheese to the wet egg mixture, stirring to separate vegetable strands.
Add the dry flour mixture and fold gently with a spatula just until incorporated and no dry pockets remain. The batter will be thick.
Divide batter evenly among the 12 muffin cups (about 3/4 full). Sprinkle the remaining 1/4 cup of cheddar cheese evenly over the muffin tops.
Bake for 22 to 25 minutes, or until the tops are golden brown, the cheese is bubbling, and a toothpick inserted into the center comes out clean.
Allow muffins to cool in the pan for 5 minutes, then transfer to a wire cooling rack. Serve warm with butter.