Start by shredding zucchini on the fine setting of a mandoline. I have not tried making this using a grater, but I'm thinking if you don't have a mandoline you could use a grater box, just making sure to grate the zucchini lengthwise so you get long ribbons.
Heat a saucepan of canola oil (about 2-3 inches deep) over medium high heat. You know it's ready (about 350 degrees) when you dangle in a zucchini strip and it sizzles. Don't put the zucchini into cold or cool oil or it will just absorb all of the oil. When the oil is hot, carefully add a good handful of the shredded zucchini to the oil - it will bubble fiercely because of the water in the zucchini, so just let it go awhile.
When the water has started cooking out (which takes a surprising amount of time, like 5 minutes or more), the bubbles will die down a bit and you will see the zucchini start to turn brown. Keep cooking it until it's mostly browned, but not burning. Remove with a spider or slotted spoon and drain on paper towels while you cook the next batch. Make sure to salt the fried shoestring zucchini while it's still hot - and since salty foods are a perfect pairing with Champagne, why not enjoy this as an appetizer with a glass of bubbly the next time you have friends over?