Clean mushrooms with a clean cloth. Remove stems and chop them finely. Brush mushroom tops with olive oil with a brush.
Saute chopped garlic and green onion in the butter. Add parsley, oregano, chopped stems, salt and pepper. Remove from heat and add bread crumbs, 1/4 cup wine and mix well.
Stir in the 1/2 cup Parmesan and 1/4 cup of the Fontina cheese to moist enough to stick together. (Add more wine if needed).
Stuff mushroom caps quite high and top with the rest of 1/4 cup of fontina cheese. Place them in a 12x14 baking dish slightly oiled.
Take remaining 1/2 cup wine and pour into bottom of baking dish, (it will run off to make a sauce and it is OK to drizzle some extra wine over the top of caps, if you like).
Bake in pre-heated 350°F oven for about 10 minutes or until cheese melts and mushrooms are cooked through. Spoon pan liquid over mushrooms and serve.
Note: these are great at room temperature, or even still oven warm. Enjoy.
MY PERSONAL UPDATED NOTE: I made this recipe again according with the last review to make the necessary corrections for the wine sauce and adding more cheese to make the sauce thicker and better. Hope this helps to improve this recipe.