Preheat the oven to 375 degrees.
Press the crust into a tart or pie pan and trim the excess crust from the edges. Bake it for 10 - 15 minutes until it is very lightly browned. Remove it from the oven.
Meanwhile, in a large bowl, whisk together the eggs, yogurt, 1/4 cup basil, salt, pepper, and Swiss cheese. Pour the mixture into the pie shell.
Top it evenly with the tomatoes and remaining basil. Sprinkle the Parmesan cheese on top and bake it for 40 - 45 minutes or until it is firm and lightly browned.
Allow the quiche to cool for at least 10 minutes before cutting it into wedges to serve, or refrigerate it for up to 3 days, or wrap it tightly and freeze it for up to 3 months.