Heat butter and olive oil in a large sauté pan over medium-low heat.
Add sliced onions to the pan and sauté, stirring frequently, until the onions are soft, golden brown, and slightly caramelized.
Add the verdolagas to the pan and use a pair of tongs to toss them with the caramelized onions. Sprinkle with salt and pepper and cover the pan with a fitted lid. Allow the verdolagas to cook for 5 minutes. Sprinkle parmesan cheese over the verdolagas and replace the lid on the pan. Allow the cheese to melt. Serve with extra parmesan and pepper.