Preheat the oven to 400 degrees F.
Slice the zucchini lengthwise in half and cut each half into very thin half moon shapes. Melt the butter in a large (preferably nonstick) skillet over medium-high heat. Saute the onion and then add the zucchini. Season with salt and pepper, to taste, and cook just until slightly tender, about 3 to 4 minutes. Spread the onion and zucchini evenly over the bottom of the pan and top with the beaten eggs. Season with salt and pepper, to taste, and allow it to cook until just starting to set at edges. Crumble the goat cheese evenly over the top, and put it in the oven until golden on top, about 3 to 5 minutes. Slightly cool the frittata in the pan before slicing. Serve it from the pan or invert it onto a plate, then slice and serve.