Trim ends off the squash and slice very thinly, about ¼-inch thick. Transfer to a colander, sprinkle with the salt, and set aside to drain for 15 to 20 minutes; rinse and pat dry.
In a large skillet, heat 1 tablespoon of the oil over medium heat. Add the squash and sauté until crisp-tender, 4 to 5 minutes. Transfer to a plate lined with paper towels to drain. Add 1 teaspoon of the oil to the skillet and sauté the shallots and garlic for 1 minute. Add the tomatoes and cook until heated through; transfer to a plate and set aside.
In a large bowl, whisk the eggs with the Parmesan and pepper. Heat the remaining 5 teaspoons oil in the skillet over medium heat, spread the squash over the pan bottom, and sprinkle with the oregano. Pour in the egg mixture and cook until set, about 5 minutes. Meanwhile, preheat the broiler. Top the frittata with the tomato mixture, Gruyère, basil and parsley. Slip under the broiler just to melt the cheese, about 1 minute.
Cut into wedges and serve warm or at room temperature.
Notes
To seed a tomato, cut it in half crosswise, then lightly squeeze and shake it to dislodge the seeds. Use a finger, if needed, to help dislodge the seed sacs.