Cook the Potatoes: Place the peeled and cubed potatoes in a large pot and cover with cold water. Add a pinch of salt and bring to a boil. Cook until the potatoes are tender, about 15-20 minutes. Drain and set aside.

Caramelize the Onions: In a large skillet, heat the butter and olive oil over medium heat. Add the sliced onions and cook, stirring occasionally, until they are caramelized and golden brown, about 15-20 minutes. If the onions start to brown too quickly, reduce the heat.

Add Broth: Once the onions are caramelized, pour in the beef broth and cook for an additional 2-3 minutes, allowing the broth to reduce slightly.

Mash the Potatoes: In a large bowl, combine the cooked potatoes and the heavy cream. Mash the potatoes until smooth and creamy. You can use a potato masher or a hand mixer for a creamier texture.

Combine: Stir the caramelized onions and broth mixture into the mashed potatoes. Add the shredded Gruyère cheese, and mix until the cheese is melted and everything is well combined. Season with salt and black pepper to taste.

Serve: Transfer the French onion mashed potatoes to a serving dish and garnish with chopped chives or parsley if desired.