In a medium bowl, whisk together the eggs and 1 teaspoon salt. In a nonstick 12-inch skillet over medium, melt 1 tablespoon of butter. Add the shallots and ½ teaspoon salt. Cover and cook, stirring occasionally, until lightly browned, about 5 minutes. Add the zucchini and cook, stirring often, until quite tender, 4 to 6 minutes. Transfer to a medium bowl and wipe out the pan. Add ½ teaspoon pepper and the goat cheese to the zucchini and stir until the cheese softens and the mixture is well combined; set aside.