Preheat the oven to 400 degrees and line a baking sheet with aluminum foil.
Brush the mushrooms of any dirt, then gently pop out the stems. Set the caps aside, then chop up the stems until finely diced.
Heat a tablespoon of olive oil in a large skillet over medium heat. Add the mushroom stems, diced onion, and minced garlic and saute until the onion is barely transparent. Add the chopped spinach and cook a moment longer, or until the spinach is wilted. Set aside to cool.
In a medium bowl, combine mushroom mixture with the cheeses, breadcrumbs, beaten egg, and Greek seasoning.
Lightly brush the lined baking sheet with olive oil, then evenly arrange the mushroom caps. Lightly brush the tops of the mushroom caps with olive oil, then season with salt and pepper. Fill each mushroom cap with a generous spoonful of the mixture.
Once all the mushroom caps are filled, transfer the baking sheet to the oven and bake for 15 - 20 minutes, or until the stuffing is golden brown and fragrant.
Note: Don't worry if the mushrooms "leak" a little while they're baking. It should dry up before they are done, but if not, transfer the caps to a few layers of paper towel to absorb any excess liquid before serving.