Beat the eggs and add a pinch of salt.
To an ovenproof pan over medium-high heat, add the olive oil and sliced mushrooms with salt and pepper and sauté them for about 5 minutes. Stir from time to time.
Lower the heat to medium, and add the finely chopped red onion, dried oregano, and butter, and cook for about 8 minutes. Stir from time to time.
Add the minced garlic and saute for about 30-40 seconds.
Add the beaten eggs (make sure the liquid from the mushrooms has mostly evaporated, depending on the type of mushrooms you have, you may need to cook them a bit more or less).
Add the goat cheese and (optionally) the finely chopped parsley.
Bake it in the preheated oven at 390°F (200°C) until cooked through, golden, and puffed up, about 15-18 minutes.