Fit the piecrust into a 9-inch deep-dish pie plate according to the package directions; fold edges under and crimp. Prick bottom and sides of the pie crust with a fork.
Line the pastry with aluminum foil and fill with pie weights or dried beans. Bake for 7 minutes. Remove the weights and foil and bake until the bottom is golden brown, 3 to 4 more minutes. Reduce the oven temperature to 400 degrees F.
While the pie crust is cooking, heat the oil in a medium skillet over medium-high heat. Add the green onions and cook 2 minutes, stirring occasionally, or until tender. Stir in the ham and garlic and cook 1 minute longer until the garlic is fragrant. Let cool slightly.
Whisk together the half-and-half, Dijon mustard, soy sauce, salt, pepper, whole eggs and egg white in a medium mixing bowl. Then add the roasted red peppers, oregano and basil.
Sprinkle the Swiss cheese over the bottom of the pastry shell. Top with 1/2 cup of the Parmesan cheese and the ham mixture. Pour the egg mixture over the top. Sprinkle with the remaining 2 tablespoons Parmesan cheese.
Bake until the center is set, 35 to 40 minutes, shielding the crust with aluminum foil, if necessary. Transfer to a wire rack and let cool for 10 minutes before serving.
Tip: If your quiche isn't quite done but you're in a pinch to get it on the table, just stick it in the microwave for 60 seconds. This will help the eggs to set up.