Heat a 9-inch cast-iron skillet over medium heat. Add the oil to the pan and warm. Add the sweet potato, onion and 1/4 teaspoon salt. Cook, stirring often, until the onions and potatoes are beginning to brown and the potatoes are cooked through, about 8 minutes. Add the kale and season with 1/2 teaspoon salt. Cook, stirring often, until the kale is wilted and tender, about 3 minutes.
Season the eggs with the remaining 1/4 teaspoon salt and stir in 1 cup of the mozzarella. Pour the egg mixture over the potato mixture. Using a rubber spatula, gently stir the mixture, making sure all the potatoes are coated in the egg and the ingredients are evenly distributed. Cook on the stove for about 3 minutes to set the bottom. Sprinkle with the remaining 1/4 cup mozzarella. Place the pan under the broiler and broil until golden brown and set, about 3 minutes. Allow to cool for 5 minutes in the pan. Run a thin knife along the edges and cut into 16 squares. Serve warm or at room temperature.