Preheat oven to 425 degrees F. Lightly coat a muffin tin with baking spray or line with paper cups. (This recipe usually yields 16 regular-sized muffins for me.)
Grate zucchini on the side of a box grater, then place over a double layer of paper towels. Set aside.
In a large bowl, whisk together the eggs and sugar. Whisk in the pumpkin puree, followed by the oil and vanilla. Stir until very smooth.
Sprinkle flour, pumpkin pie spice, baking soda, and salt on top. Fold just until evenly combined.
Wrap the paper towels tightly around the shredded zucchini, hold over the sink, and press all over with your hands to squeeze out excess water. Once you’re getting only small drips, fold back the paper towels and shake the zucchini out over the batter. Fold again gently until evenly mixed.
Divide batter evenly among muffin cups, filling each nearly to the top. (Use a large dough scoop, or you can fill them with a scant 1/3 cup each.) If desired, sprinkle a tiny amount of turbinado sugar or brown sugar on top of each muffin.
Bake at 425 degrees for 5 minutes, then, without opening the oven door, reduce the temperature to 350 degrees and bake for another 12-15 minutes. When done, a toothpick inserted into the centers should come out with moist crumbs, but no large streaks of batter.
Let cool for 10-15 minutes in the pan, then remove to a wire rack to cool completely.