Preheat the oven to 400F degrees.
Bring the quinoa and water to a boil in a small saucepan. Cover, reduce to a simmer and cook for 10 minutes. Turn the heat off and leave the covered saucepan on the burner for another 4 minutes. Fluff with a fork and allow the quinoa to cool.
Gently twist the mushroom stems to remove them. Finely dice the stems. Melt the butter or olive oil in a small saucepan and sauté the stems and onion on medium heat until the onion is opaque and tender, 3 to 5 minutes. Remove from the heat and allow to cool.
Stir the quinoa, pistachios, parsley, oregano, pepper sauce, salt and 2 tablespoons of freshly grated Parmesan cheese into the mushroom and onion mixture. Using a teaspoon, scoop the filling into the mushroom caps and place them on a baking sheet. (They can be refrigerated at this point for up to 24 hours.)
Bake for 13-15 minutes, depending on their size, until the mushrooms are tender. Sprinkle with remaining Parmesan cheese. Add a dash of red pepper flakes, if desired. Serve warm.