Cut the eggs lengthwise in half. Arrange the halves cut-side-down in the prepared baking dish.
Pour the half-and-half into a heavy saucepan, and place it over medium heat. When the liquid begins to bubble, whisk in the cornstarch mixture. Whisk vigorously for one full minute while the sauce simmers and thickens. Remove from heat.
Whisk in the Swiss cheese, mustard, nutmeg, and salt. Continue whisking until the cheese melts. Pour the sauce over the eggs, and sprinkle on the grated Asiago or Parmesan cheese.
If you are not going to bake right away, allow the dish to cool to room temperature. Then cover with cling film and refrigerate for 24-48 hours.
Center the oven rack, and preheat the oven to 350°F. Bake until the sauce bubbles and the top turns golden brown -- 25-30 minutes. Garnish with the chopped chives or minced parsley.