I make this recipe quite often. It goes together rather quickly, and it's a great as a weekend breakfast dish or even for a special brunch. It's not really a French dish, but this site doesn't have Switzerland as a cuisine choice, so I selected French since it was the closest cuisine choice.
Preheat oven to 400°F. Butter an 8x8-inch baking dish; set aside.
Melt the butter in a medium sauté pan over medium-high heat. When the butter is foamy, add the mushrooms and season lightly with salt and pepper. Cook until tender, about 7 minutes, stirring occasionally. Stir in the garlic and thyme during the last 2 minutes of cooking.
Scrape the mushrooms into the prepared baking dish.
Crack the eggs over the mushrooms (if you prefer solid yolks, you might prefer to break the yolks). Season eggs with salt and pepper. Sprinkle the cheese on top of the eggs, then drizzle the cream on top of the cheese.
Bake about 20-12 minutes or until the whites are set and the yolks are velvety and starting to set (or bake another 2-3 minutes for a more solid yolk). Serve immediately.