Preheat oven to 375° F. In large bowl, whisk together cream, eggs, flour, salt, pepper and nutmeg. Stir in chicken, broccoli, sun-dried tomatoes and green onions; set aside. Sprinkle cheese evenly into bottom of each pie shell; place pie shells on parchment paper–lined baking sheet.
Divide chicken mixture between pie shells; immediately bake in bottom third of oven for 30 minutes or until filling is set but still jiggles slightly in center and pastry is golden. Let stand for 10 minutes before slicing into wedges.
Tip: Switch up the flavor by using Cheddar cheese instead of Swiss cheese.