Make the vinaigrette: In a bowl, combine the oil and vinegar. Stir well to combine. Set aside.
Make the frittata: Melt the butter in a nonstick skillet over medium-high heat, then add and saute the mushrooms until lightly browned. Remove from the heat and reserve the melted butter to grease a baking dish.
On a plate, combine the green onions with fried mushrooms, then sprinkle with half the salt and pepper. Fold in the parsley.
In another bowl, whisk together the eggs, cheese, mayo, and remaining salt and pepper.
Add the mushroom mixture to the egg mixture. Stir to combine well.
Pour the mixture into the greased baking dish. Bake for about 40 minutes until lightly browned and puffed.