Preheat oven to 325 degrees. Using the larger side of a box grater, grate the zucchini. Using several paper towels, place the zucchini in the center and wring the zucchini to remove excess moisture. In an 8 inch cast iron skillet melt the butter. Add the zucchini and saute over medium heat until the moisture evaporates and zucchini is tender.
In a medium bowl whisk the eggs and heavy cream. Stir in the ham, cheddar cheese, salt and pepper and add the zucchini.
Pour the mixture into the skillet and bake for 20-25 minutes until it is set.