Place the grated zucchini in a colander and sprinkle with ½ teaspoon salt. Mix the salt thoroughly into the zucchini with your hands and let it sit for 10 minutes. Grate the cheese. Preheat the oven to 350°F and position the rack in the middle. Butter a 9 to 10-inch pie plate or quiche dish.
Rinse the zucchini under running water very thoroughly. Using clean hands, squeeze out as much liquid from the zucchini as possible. Transfer the zucchini to a medium bowl and mix in the dried dill and fresh chives. In a separate medium bowl, combine the remaining ¼ teaspoon salt, white pepper, eggs, and heavy cream. Beat the mixture until frothy.
Layer one-quarter of the grated cheese evenly in the prepared pie plate. Add one-third of the zucchini mixture on top of the cheese. Continue layering alternating between cheese and zucchini until all is used. Pour the egg and cream mixture evenly over the layers.
Place the quiche in the oven and bake for 40 to 50 minutes, or until the top is browned and the center is just set but still slightly soft. Remove from the oven and allow to cool to room temperature before serving.
Store any leftovers covered with cling film in the refrigerator. Reheat in a microwave or oven. This dish is delicious served warm alongside a green salad. Serves 6, with approximately 4.1g net carbs per serving.