How to Make It
Preheat the broiler and position a rack 8 inches from the heat. In a medium skillet, heat 2 tablespoons of the olive oil. Add the mushrooms and shallot, season with salt and pepper and cook over high heat, stirring occasionally, until browned, about 8 minutes. Add the stock and simmer until reduced by half, about 5 minutes. Stir in the vinegar and 1 tablespoon of the butter and keep warm.
In a medium bowl, whisk the eggs with the milk and a pinch of salt and pepper. In a large nonstick skillet, melt the remaining 1 tablespoon of butter in the remaining 1 tablespoon of oil. Add the eggs and cook over moderate heat, stirring gently, until the bottom is set but the eggs are still soft in the center, 3 to 4 minutes. Using a rubber spatula, fold the omelet into thirds and slide onto a heatproof platter.
Pour the mushrooms over the omelet and top with the cheese slices. Broil for about 30 seconds, until the cheese is melted and bubbling. Sprinkle with the parsley and serve right away.