In a skillet saute mushrooms and onion in oil over medium heat until mushrooms release their liquid and onions are translucent. Do not discard resulting liquid.
In large bowl, beat eggs, salt, pepper, basil, oregano, and sweet rice flour. Add mushroom and onion mixture (including any liquid) and remaining ingredients to eggs. Mix.
Pour into a well oiled pie plate, or line the pie plate with oiled parchment. Bake at 425 degrees F for 10 minutes. Reduce oven temperature to 350 degrees and bake for an additional 25 to 35 minutes. Let quiche stand for a few minutes before cutting and serving to ensure quiche sets up properly.
To freeze: Allow quiche to cool. Wrap well for the freezer, removing all air, seal, label and freeze.