In a large skillet, melt the butter and olive oil together over medium heat. Add the onions and saute for 5-6 minutes, stirring occasionally.
Add salt, pepper and oregano. Give it a good stir. Cover the skillet and cook until the onions are soft and wonderfully golden, stirring occasionally.
Add the beef broth and cook covered until onions are golden brown. Scrape any pieces of onion that stick to the bottom of the skillet and remove from heat.
Preheat a large skillet over medium heat. Butter the slices of bread on one side. Place a slice of bread on the skillet, buttered side down. Sprinkle on a bit of the shredded cheese.
Place a generous layer of the onion mixture on top.
Then top with more cheese and put a second slice of bread on top, buttered side up.
Flip when the bottom is crispy and golden and cheese is melted. Repeat on second side.