Wash and remove both ends of the zucchini.
Using a food processor or box grater, shred the zucchini. Note: use the large grater openings.
Line a colander with a clean dish towel and place shredded zucchini into the lined colander and set aside.
Once shredded the zucchini will begin releasing its juices. As it does its juices will be absorbed into the towel. You may also need to squeeze additional juices from the shredded zucchini. To do this tightly twist the clean dish towel around the zucchini to wring more juices from the zucchini.
Heat EVOO and butter in a large skillet over medium heat.
Once heated add the finely chopped onions and saute until they become softened and translucent, about 3-5 minutes
Add garlic and oregano and allow to saute a minute or two more.
Squeeze juices from zucchini one more time and add them into the pan.
Season with 1 teaspoon of salt and pepper to taste, mixing the ingredients together gently, but thoroughly.
Saute the zucchini for another 5 minutes or until tender, but not mushy.
Remove from heat and add the grated cheese and gently combine.
Spoon into a serving bowl, top with chopped basil, and enjoy.