Bring a large pot of salted water to a boil. Cook the potato gnocchi according to the package directions until they float to the surface, indicating they are done. Drain thoroughly and set aside.

In a large, heavy-bottomed skillet, heat the butter and olive oil over medium-high heat until melted and shimmering. Add the sliced onions and stir to coat them evenly in the fat. Sauté for about 5 minutes, stirring occasionally, until the onions begin to soften and develop a golden hue.

Add the sliced mushrooms to the pan and toss them with the onions. Season with a pinch of salt to encourage the mushrooms to release their moisture. Continue to cook for approximately 10 minutes, stirring occasionally, until the mushrooms are tender, golden brown around the edges, and most of their liquid has evaporated.

Pour in the white wine or broth, and use a wooden spoon to deglaze the skillet, scraping up any browned bits stuck to the bottom. Let the liquid simmer and reduce by half, allowing the flavors to concentrate.

Add the cooked gnocchi to the skillet and gently toss everything together until the gnocchi is well-coated and warmed through. Taste and adjust seasoning with additional salt and freshly cracked black pepper as needed.

To serve, spoon a generous portion of the gnocchi mixture into a bowl or plate and finish with a generous sprinkle of freshly grated parmesan cheese. Serve hot.